OCMEBilingual mathematics & derivations繁體中文

Original Open Covers, Indexed Subcovers and Inclusion-Minimality Boundaries

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations · 2026-10-14 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

1. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation

([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-audit · candidate

2. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 1], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation

([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-audit · candidate

3. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation

([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-audit · candidate

4. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 1], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation

([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-audit · candidate

5. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 1], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation

([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-audit · candidate

6. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 1], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation

([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-audit · candidate

7. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation

([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-audit · candidate

8. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation

([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-audit · candidate

9. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation

([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-audit · candidate

10. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 2], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation

([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-audit · candidate

11. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation

([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-audit · candidate

12. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 2], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation

([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-audit · candidate

13. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation

([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-audit · candidate

14. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation

([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-audit · candidate

15. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation

([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-audit · candidate

16. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation

([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-audit · candidate

17. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation

([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-audit · candidate

18. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation

([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-audit · candidate

19. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation

([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-audit · candidate

20. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation

([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-audit · candidate

21. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation

([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-audit · candidate

22. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation

([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-audit · candidate

23. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation

([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-audit · candidate

24. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation

([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-audit · candidate

25. Coverage and Openness Are Independent Original-Family Obligations

Use the complete finite topology=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.

View answer and explanation

([], [0, 1, 2, 3, 4], Yes, Yes)

List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-audit · candidate

26. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation

([1], [[0, 1, 2, 3]], [0, 1, 2, 3], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 2, 3]], [0, 1, 2, 3], Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate

27. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 1], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation

([1], [[0, 1, 2, 4]], [0, 1, 2, 4], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 2, 4]], [0, 1, 2, 4], Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate

28. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation

([1], [[0, 1, 2, 3]], [0, 1, 2, 3], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 2, 3]], [0, 1, 2, 3], Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate

29. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 1], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation

([1], [[0, 1, 3, 4]], [0, 1, 3, 4], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 3, 4]], [0, 1, 3, 4], Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate

30. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 1], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation

([1], [[0, 1, 2, 4]], [0, 1, 2, 4], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 2, 4]], [0, 1, 2, 4], Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate

31. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 1], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation

([1], [[0, 1, 3, 4]], [0, 1, 3, 4], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 3, 4]], [0, 1, 3, 4], Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate

32. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation

([1], [[0, 1, 2, 3]], [0, 1, 2, 3], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 2, 3]], [0, 1, 2, 3], Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate

33. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation

([1], [[0, 1, 2, 4]], [0, 1, 2, 4], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 2, 4]], [0, 1, 2, 4], Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate

34. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation

([1], [[0, 1, 2, 3]], [0, 1, 2, 3], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 2, 3]], [0, 1, 2, 3], Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate

35. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 2], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation

([1], [[0, 2, 3, 4]], [0, 2, 3, 4], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 2, 3, 4]], [0, 2, 3, 4], Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate

36. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation

([1], [[0, 1, 2, 4]], [0, 1, 2, 4], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 2, 4]], [0, 1, 2, 4], Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate

37. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 2], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation

([1], [[0, 2, 3, 4]], [0, 2, 3, 4], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 2, 3, 4]], [0, 2, 3, 4], Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate

38. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation

([1], [[0, 1, 2, 3]], [0, 1, 2, 3], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 2, 3]], [0, 1, 2, 3], Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate

39. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation

([1], [[0, 1, 3, 4]], [0, 1, 3, 4], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 3, 4]], [0, 1, 3, 4], Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate

40. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation

([1], [[0, 1, 2, 3]], [0, 1, 2, 3], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 2, 3]], [0, 1, 2, 3], Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate

41. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation

([1], [[0, 2, 3, 4]], [0, 2, 3, 4], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 2, 3, 4]], [0, 2, 3, 4], Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate

42. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation

([1], [[0, 1, 3, 4]], [0, 1, 3, 4], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 3, 4]], [0, 1, 3, 4], Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate

43. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation

([1], [[0, 2, 3, 4]], [0, 2, 3, 4], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 2, 3, 4]], [0, 2, 3, 4], Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate

44. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation

([1], [[0, 1, 2, 4]], [0, 1, 2, 4], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 2, 4]], [0, 1, 2, 4], Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate

45. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation

([1], [[0, 1, 3, 4]], [0, 1, 3, 4], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 3, 4]], [0, 1, 3, 4], Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate

46. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation

([1], [[0, 1, 2, 4]], [0, 1, 2, 4], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 2, 4]], [0, 1, 2, 4], Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate

47. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation

([1], [[0, 2, 3, 4]], [0, 2, 3, 4], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 2, 3, 4]], [0, 2, 3, 4], Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate

48. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation

([1], [[0, 1, 3, 4]], [0, 1, 3, 4], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 3, 4]], [0, 1, 3, 4], Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate

49. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation

([1], [[0, 2, 3, 4]], [0, 2, 3, 4], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 2, 3, 4]], [0, 2, 3, 4], Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate

50. A Certified Subcover Selects Only from the Qualified Original Cover

Use the complete finite topology=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).

View answer and explanation

([1], [[0, 1, 2, 3]], [0, 1, 2, 3], Yes)

Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 2, 3]], [0, 1, 2, 3], Yes).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-subcover-audit · candidate

51. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation

([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate

52. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 1], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation

([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate

53. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation

([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate

54. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 1], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation

([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate

55. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 1], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation

([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate

56. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 1], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation

([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate

57. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation

([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate

58. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation

([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate

59. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation

([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate

60. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 2], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation

([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate

61. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation

([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate

62. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 2], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation

([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate

63. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation

([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate

64. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation

([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate

65. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation

([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate

66. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation

([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate

67. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation

([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate

68. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation

([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate

69. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation

([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate

70. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation

([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate

71. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation

([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate

72. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation

([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate

73. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation

([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate

74. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation

([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate

75. Inclusion-Minimal Does Not Mean Minimum Cardinality

Use the complete finite topology=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.

View answer and explanation

([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)

Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-minimal-subcover · candidate

76. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation

([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate

77. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 1], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation

([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate

78. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation

([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate

79. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 1], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation

([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate

80. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 1], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation

([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate

81. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 1], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation

([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate

82. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation

([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate

83. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation

([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate

84. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation

([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate

85. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 2], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation

([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate

86. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation

([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate

87. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 2], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation

([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate

88. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation

([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate

89. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation

([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate

90. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation

([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate

91. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation

([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate

92. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation

([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate

93. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation

([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate

94. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation

([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate

95. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation

([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate

96. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation

([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate

97. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation

([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate

98. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation

([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate

99. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation

([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate

100. One Supplied Finite Subcover Is Not the Universal Compactness Definition

Use the complete finite topology=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).

View answer and explanation

([1], Yes, No, one_finite_subcover_does_not_prove_compactness)

The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness).

Reviewed step-by-step derivation

Open cover and subcover · qf-mod32-cover-single-cover-boundary · candidate

Canonical question JSONL · English JSONL · Validation report