OCMEBilingual mathematics & derivations繁體中文

Inverse Existence, Whole-Table Types, and Two-Sided Identities

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-20261012-advance-20261017-r02-inverse-function-foundations · 2026-10-12 · 100 questions

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

1. Bijection Guard before Constructing Both Inverse Compositions

Use the complete total map Fin(0)=∅→Fin(0) with images=[] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, bijectivity, inverse table or not-an-inverse-function, g∘f values or not-evaluated, f∘g values or not-evaluated). Do not confuse inverse functions with reciprocal function values.

View answer and explanation

([], Yes, [], [], [])

Require a unique preimage for every declared target before constructing the inverse; compute the two ordered compositions separately. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], Yes, [], [], []).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-bijection-gate-and-table · candidate

2. Bijection Guard before Constructing Both Inverse Compositions

Use the complete total map Fin(0)=∅→Fin(2) with images=[] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, bijectivity, inverse table or not-an-inverse-function, g∘f values or not-evaluated, f∘g values or not-evaluated). Do not confuse inverse functions with reciprocal function values.

View answer and explanation

([], No, not-an-inverse-function, not-evaluated, not-evaluated)

Require a unique preimage for every declared target before constructing the inverse; compute the two ordered compositions separately. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], No, not-an-inverse-function, not-evaluated, not-evaluated).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-bijection-gate-and-table · candidate

3. Bijection Guard before Constructing Both Inverse Compositions

Use the complete total map Fin(1)={0,…,0}→Fin(1) with images=[0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, bijectivity, inverse table or not-an-inverse-function, g∘f values or not-evaluated, f∘g values or not-evaluated). Do not confuse inverse functions with reciprocal function values.

View answer and explanation

([0], Yes, [0], [0], [0])

Require a unique preimage for every declared target before constructing the inverse; compute the two ordered compositions separately. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0], Yes, [0], [0], [0]).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-bijection-gate-and-table · candidate

4. Bijection Guard before Constructing Both Inverse Compositions

Use the complete total map Fin(3)={0,…,2}→Fin(3) with images=[0, 1, 2] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, bijectivity, inverse table or not-an-inverse-function, g∘f values or not-evaluated, f∘g values or not-evaluated). Do not confuse inverse functions with reciprocal function values.

View answer and explanation

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

Require a unique preimage for every declared target before constructing the inverse; compute the two ordered compositions separately. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2], Yes, [0, 1, 2], [0, 1, 2], [0, 1, 2]).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-bijection-gate-and-table · candidate

5. Bijection Guard before Constructing Both Inverse Compositions

Use the complete total map Fin(3)={0,…,2}→Fin(3) with images=[0, 0, 1] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, bijectivity, inverse table or not-an-inverse-function, g∘f values or not-evaluated, f∘g values or not-evaluated). Do not confuse inverse functions with reciprocal function values.

View answer and explanation

([0, 0, 1], No, not-an-inverse-function, not-evaluated, not-evaluated)

Require a unique preimage for every declared target before constructing the inverse; compute the two ordered compositions separately. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1], No, not-an-inverse-function, not-evaluated, not-evaluated).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-bijection-gate-and-table · candidate

6. Bijection Guard before Constructing Both Inverse Compositions

Use the complete total map Fin(3)={0,…,2}→Fin(2) with images=[0, 1, 1] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, bijectivity, inverse table or not-an-inverse-function, g∘f values or not-evaluated, f∘g values or not-evaluated). Do not confuse inverse functions with reciprocal function values.

View answer and explanation

([0, 1, 1], No, not-an-inverse-function, not-evaluated, not-evaluated)

Require a unique preimage for every declared target before constructing the inverse; compute the two ordered compositions separately. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 1], No, not-an-inverse-function, not-evaluated, not-evaluated).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-bijection-gate-and-table · candidate

7. Bijection Guard before Constructing Both Inverse Compositions

Use the complete total map Fin(2)={0,…,1}→Fin(3) with images=[0, 1] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, bijectivity, inverse table or not-an-inverse-function, g∘f values or not-evaluated, f∘g values or not-evaluated). Do not confuse inverse functions with reciprocal function values.

View answer and explanation

([0, 1], No, not-an-inverse-function, not-evaluated, not-evaluated)

Require a unique preimage for every declared target before constructing the inverse; compute the two ordered compositions separately. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1], No, not-an-inverse-function, not-evaluated, not-evaluated).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-bijection-gate-and-table · candidate

8. Bijection Guard before Constructing Both Inverse Compositions

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 0, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, bijectivity, inverse table or not-an-inverse-function, g∘f values or not-evaluated, f∘g values or not-evaluated). Do not confuse inverse functions with reciprocal function values.

View answer and explanation

([0, 0, 0, 0, 0], No, not-an-inverse-function, not-evaluated, not-evaluated)

Require a unique preimage for every declared target before constructing the inverse; compute the two ordered compositions separately. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 0, 0, 0], No, not-an-inverse-function, not-evaluated, not-evaluated).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-bijection-gate-and-table · candidate

9. Bijection Guard before Constructing Both Inverse Compositions

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 0, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, bijectivity, inverse table or not-an-inverse-function, g∘f values or not-evaluated, f∘g values or not-evaluated). Do not confuse inverse functions with reciprocal function values.

View answer and explanation

([1, 0, 0, 0, 0], No, not-an-inverse-function, not-evaluated, not-evaluated)

Require a unique preimage for every declared target before constructing the inverse; compute the two ordered compositions separately. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 0, 0, 0], No, not-an-inverse-function, not-evaluated, not-evaluated).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-bijection-gate-and-table · candidate

10. Bijection Guard before Constructing Both Inverse Compositions

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 0, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, bijectivity, inverse table or not-an-inverse-function, g∘f values or not-evaluated, f∘g values or not-evaluated). Do not confuse inverse functions with reciprocal function values.

View answer and explanation

([2, 0, 0, 0, 0], No, not-an-inverse-function, not-evaluated, not-evaluated)

Require a unique preimage for every declared target before constructing the inverse; compute the two ordered compositions separately. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 0, 0, 0], No, not-an-inverse-function, not-evaluated, not-evaluated).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-bijection-gate-and-table · candidate

11. Bijection Guard before Constructing Both Inverse Compositions

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 1, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, bijectivity, inverse table or not-an-inverse-function, g∘f values or not-evaluated, f∘g values or not-evaluated). Do not confuse inverse functions with reciprocal function values.

View answer and explanation

([0, 1, 0, 0, 0], No, not-an-inverse-function, not-evaluated, not-evaluated)

Require a unique preimage for every declared target before constructing the inverse; compute the two ordered compositions separately. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 0, 0, 0], No, not-an-inverse-function, not-evaluated, not-evaluated).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-bijection-gate-and-table · candidate

12. Bijection Guard before Constructing Both Inverse Compositions

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 1, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, bijectivity, inverse table or not-an-inverse-function, g∘f values or not-evaluated, f∘g values or not-evaluated). Do not confuse inverse functions with reciprocal function values.

View answer and explanation

([1, 1, 0, 0, 0], No, not-an-inverse-function, not-evaluated, not-evaluated)

Require a unique preimage for every declared target before constructing the inverse; compute the two ordered compositions separately. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 0, 0, 0], No, not-an-inverse-function, not-evaluated, not-evaluated).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-bijection-gate-and-table · candidate

13. Bijection Guard before Constructing Both Inverse Compositions

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 1, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, bijectivity, inverse table or not-an-inverse-function, g∘f values or not-evaluated, f∘g values or not-evaluated). Do not confuse inverse functions with reciprocal function values.

View answer and explanation

([2, 1, 0, 0, 0], No, not-an-inverse-function, not-evaluated, not-evaluated)

Require a unique preimage for every declared target before constructing the inverse; compute the two ordered compositions separately. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 0, 0, 0], No, not-an-inverse-function, not-evaluated, not-evaluated).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-bijection-gate-and-table · candidate

14. Bijection Guard before Constructing Both Inverse Compositions

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 2, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, bijectivity, inverse table or not-an-inverse-function, g∘f values or not-evaluated, f∘g values or not-evaluated). Do not confuse inverse functions with reciprocal function values.

View answer and explanation

([0, 2, 0, 0, 0], No, not-an-inverse-function, not-evaluated, not-evaluated)

Require a unique preimage for every declared target before constructing the inverse; compute the two ordered compositions separately. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 0, 0, 0], No, not-an-inverse-function, not-evaluated, not-evaluated).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-bijection-gate-and-table · candidate

15. Bijection Guard before Constructing Both Inverse Compositions

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 2, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, bijectivity, inverse table or not-an-inverse-function, g∘f values or not-evaluated, f∘g values or not-evaluated). Do not confuse inverse functions with reciprocal function values.

View answer and explanation

([1, 2, 0, 0, 0], No, not-an-inverse-function, not-evaluated, not-evaluated)

Require a unique preimage for every declared target before constructing the inverse; compute the two ordered compositions separately. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 0, 0, 0], No, not-an-inverse-function, not-evaluated, not-evaluated).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-bijection-gate-and-table · candidate

16. Bijection Guard before Constructing Both Inverse Compositions

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 2, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, bijectivity, inverse table or not-an-inverse-function, g∘f values or not-evaluated, f∘g values or not-evaluated). Do not confuse inverse functions with reciprocal function values.

View answer and explanation

([2, 2, 0, 0, 0], No, not-an-inverse-function, not-evaluated, not-evaluated)

Require a unique preimage for every declared target before constructing the inverse; compute the two ordered compositions separately. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2, 0, 0, 0], No, not-an-inverse-function, not-evaluated, not-evaluated).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-bijection-gate-and-table · candidate

17. Bijection Guard before Constructing Both Inverse Compositions

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 0, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, bijectivity, inverse table or not-an-inverse-function, g∘f values or not-evaluated, f∘g values or not-evaluated). Do not confuse inverse functions with reciprocal function values.

View answer and explanation

([0, 0, 1, 0, 0], No, not-an-inverse-function, not-evaluated, not-evaluated)

Require a unique preimage for every declared target before constructing the inverse; compute the two ordered compositions separately. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1, 0, 0], No, not-an-inverse-function, not-evaluated, not-evaluated).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-bijection-gate-and-table · candidate

18. Bijection Guard before Constructing Both Inverse Compositions

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 0, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, bijectivity, inverse table or not-an-inverse-function, g∘f values or not-evaluated, f∘g values or not-evaluated). Do not confuse inverse functions with reciprocal function values.

View answer and explanation

([1, 0, 1, 0, 0], No, not-an-inverse-function, not-evaluated, not-evaluated)

Require a unique preimage for every declared target before constructing the inverse; compute the two ordered compositions separately. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 1, 0, 0], No, not-an-inverse-function, not-evaluated, not-evaluated).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-bijection-gate-and-table · candidate

19. Bijection Guard before Constructing Both Inverse Compositions

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 0, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, bijectivity, inverse table or not-an-inverse-function, g∘f values or not-evaluated, f∘g values or not-evaluated). Do not confuse inverse functions with reciprocal function values.

View answer and explanation

([2, 0, 1, 0, 0], No, not-an-inverse-function, not-evaluated, not-evaluated)

Require a unique preimage for every declared target before constructing the inverse; compute the two ordered compositions separately. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 1, 0, 0], No, not-an-inverse-function, not-evaluated, not-evaluated).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-bijection-gate-and-table · candidate

20. Bijection Guard before Constructing Both Inverse Compositions

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 1, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, bijectivity, inverse table or not-an-inverse-function, g∘f values or not-evaluated, f∘g values or not-evaluated). Do not confuse inverse functions with reciprocal function values.

View answer and explanation

([0, 1, 1, 0, 0], No, not-an-inverse-function, not-evaluated, not-evaluated)

Require a unique preimage for every declared target before constructing the inverse; compute the two ordered compositions separately. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 1, 0, 0], No, not-an-inverse-function, not-evaluated, not-evaluated).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-bijection-gate-and-table · candidate

21. Bijection Guard before Constructing Both Inverse Compositions

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 1, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, bijectivity, inverse table or not-an-inverse-function, g∘f values or not-evaluated, f∘g values or not-evaluated). Do not confuse inverse functions with reciprocal function values.

View answer and explanation

([1, 1, 1, 0, 0], No, not-an-inverse-function, not-evaluated, not-evaluated)

Require a unique preimage for every declared target before constructing the inverse; compute the two ordered compositions separately. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 1, 0, 0], No, not-an-inverse-function, not-evaluated, not-evaluated).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-bijection-gate-and-table · candidate

22. Bijection Guard before Constructing Both Inverse Compositions

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 1, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, bijectivity, inverse table or not-an-inverse-function, g∘f values or not-evaluated, f∘g values or not-evaluated). Do not confuse inverse functions with reciprocal function values.

View answer and explanation

([2, 1, 1, 0, 0], No, not-an-inverse-function, not-evaluated, not-evaluated)

Require a unique preimage for every declared target before constructing the inverse; compute the two ordered compositions separately. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 1, 0, 0], No, not-an-inverse-function, not-evaluated, not-evaluated).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-bijection-gate-and-table · candidate

23. Bijection Guard before Constructing Both Inverse Compositions

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 2, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, bijectivity, inverse table or not-an-inverse-function, g∘f values or not-evaluated, f∘g values or not-evaluated). Do not confuse inverse functions with reciprocal function values.

View answer and explanation

([0, 2, 1, 0, 0], No, not-an-inverse-function, not-evaluated, not-evaluated)

Require a unique preimage for every declared target before constructing the inverse; compute the two ordered compositions separately. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 1, 0, 0], No, not-an-inverse-function, not-evaluated, not-evaluated).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-bijection-gate-and-table · candidate

24. Bijection Guard before Constructing Both Inverse Compositions

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 2, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, bijectivity, inverse table or not-an-inverse-function, g∘f values or not-evaluated, f∘g values or not-evaluated). Do not confuse inverse functions with reciprocal function values.

View answer and explanation

([1, 2, 1, 0, 0], No, not-an-inverse-function, not-evaluated, not-evaluated)

Require a unique preimage for every declared target before constructing the inverse; compute the two ordered compositions separately. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 1, 0, 0], No, not-an-inverse-function, not-evaluated, not-evaluated).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-bijection-gate-and-table · candidate

25. Bijection Guard before Constructing Both Inverse Compositions

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 2, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, bijectivity, inverse table or not-an-inverse-function, g∘f values or not-evaluated, f∘g values or not-evaluated). Do not confuse inverse functions with reciprocal function values.

View answer and explanation

([2, 2, 1, 0, 0], No, not-an-inverse-function, not-evaluated, not-evaluated)

Require a unique preimage for every declared target before constructing the inverse; compute the two ordered compositions separately. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2, 1, 0, 0], No, not-an-inverse-function, not-evaluated, not-evaluated).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-bijection-gate-and-table · candidate

26. Whole Candidate Inverse Type Check and Two Independent Identities

Use the complete total map Fin(0)=∅→Fin(0) with images=[] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Candidate g:Fin(0)→Fin(0) has g=[]. Give (images, per-cell g type mask, whole-g eligibility, g∘f values, f∘g values, first identity?, second identity?). If any g cell is untyped, use not-evaluated in the last four fields.

View answer and explanation

([], [], Yes, [], [], Yes, Yes)

Check all candidate cells, including unused ones, before evaluating either composition. A one-sided identity does not establish the other. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [], Yes, [], [], Yes, Yes).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-candidate-two-sided · candidate

27. Whole Candidate Inverse Type Check and Two Independent Identities

Use the complete total map Fin(0)=∅→Fin(2) with images=[] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Candidate g:Fin(2)→Fin(0) has g=[-1, 0]. Give (images, per-cell g type mask, whole-g eligibility, g∘f values, f∘g values, first identity?, second identity?). If any g cell is untyped, use not-evaluated in the last four fields.

View answer and explanation

([], [No, No], No, not-evaluated, not-evaluated, not-evaluated, not-evaluated)

Check all candidate cells, including unused ones, before evaluating either composition. A one-sided identity does not establish the other. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [No, No], No, not-evaluated, not-evaluated, not-evaluated, not-evaluated).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-candidate-two-sided · candidate

28. Whole Candidate Inverse Type Check and Two Independent Identities

Use the complete total map Fin(2)={0,…,1}→Fin(3) with images=[0, 1] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Candidate g:Fin(3)→Fin(2) has g=[0, 1, 0]. Give (images, per-cell g type mask, whole-g eligibility, g∘f values, f∘g values, first identity?, second identity?). If any g cell is untyped, use not-evaluated in the last four fields.

View answer and explanation

([0, 1], [Yes, Yes, Yes], Yes, [0, 1], [0, 1, 0], Yes, No)

Check all candidate cells, including unused ones, before evaluating either composition. A one-sided identity does not establish the other. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1], [Yes, Yes, Yes], Yes, [0, 1], [0, 1, 0], Yes, No).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-candidate-two-sided · candidate

29. Whole Candidate Inverse Type Check and Two Independent Identities

Use the complete total map Fin(3)={0,…,2}→Fin(2) with images=[0, 1, 1] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Candidate g:Fin(2)→Fin(3) has g=[0, 1]. Give (images, per-cell g type mask, whole-g eligibility, g∘f values, f∘g values, first identity?, second identity?). If any g cell is untyped, use not-evaluated in the last four fields.

View answer and explanation

([0, 1, 1], [Yes, Yes], Yes, [0, 1, 1], [0, 1], No, Yes)

Check all candidate cells, including unused ones, before evaluating either composition. A one-sided identity does not establish the other. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 1], [Yes, Yes], Yes, [0, 1, 1], [0, 1], No, Yes).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-candidate-two-sided · candidate

30. Whole Candidate Inverse Type Check and Two Independent Identities

Use the complete total map Fin(2)={0,…,1}→Fin(2) with images=[1, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Candidate g:Fin(2)→Fin(2) has g=[1, 0]. Give (images, per-cell g type mask, whole-g eligibility, g∘f values, f∘g values, first identity?, second identity?). If any g cell is untyped, use not-evaluated in the last four fields.

View answer and explanation

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

Check all candidate cells, including unused ones, before evaluating either composition. A one-sided identity does not establish the other. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0], [Yes, Yes], Yes, [0, 1], [0, 1], Yes, Yes).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-candidate-two-sided · candidate

31. Whole Candidate Inverse Type Check and Two Independent Identities

Use the complete total map Fin(2)={0,…,1}→Fin(2) with images=[1, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Candidate g:Fin(2)→Fin(2) has g=[0, 1]. Give (images, per-cell g type mask, whole-g eligibility, g∘f values, f∘g values, first identity?, second identity?). If any g cell is untyped, use not-evaluated in the last four fields.

View answer and explanation

([1, 0], [Yes, Yes], Yes, [1, 0], [1, 0], No, No)

Check all candidate cells, including unused ones, before evaluating either composition. A one-sided identity does not establish the other. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0], [Yes, Yes], Yes, [1, 0], [1, 0], No, No).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-candidate-two-sided · candidate

32. Whole Candidate Inverse Type Check and Two Independent Identities

Use the complete total map Fin(2)={0,…,1}→Fin(2) with images=[0, 1] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Candidate g:Fin(2)→Fin(2) has g=[0, 2]. Give (images, per-cell g type mask, whole-g eligibility, g∘f values, f∘g values, first identity?, second identity?). If any g cell is untyped, use not-evaluated in the last four fields.

View answer and explanation

([0, 1], [Yes, No], No, not-evaluated, not-evaluated, not-evaluated, not-evaluated)

Check all candidate cells, including unused ones, before evaluating either composition. A one-sided identity does not establish the other. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1], [Yes, No], No, not-evaluated, not-evaluated, not-evaluated, not-evaluated).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-candidate-two-sided · candidate

33. Whole Candidate Inverse Type Check and Two Independent Identities

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 0, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Candidate g:Fin(3)→Fin(5) has g=[0, 2, 5]. Give (images, per-cell g type mask, whole-g eligibility, g∘f values, f∘g values, first identity?, second identity?). If any g cell is untyped, use not-evaluated in the last four fields.

View answer and explanation

([0, 0, 0, 0, 0], [Yes, Yes, No], No, not-evaluated, not-evaluated, not-evaluated, not-evaluated)

Check all candidate cells, including unused ones, before evaluating either composition. A one-sided identity does not establish the other. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 0, 0, 0], [Yes, Yes, No], No, not-evaluated, not-evaluated, not-evaluated, not-evaluated).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-candidate-two-sided · candidate

34. Whole Candidate Inverse Type Check and Two Independent Identities

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 0, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Candidate g:Fin(3)→Fin(5) has g=[0, 2, 4]. Give (images, per-cell g type mask, whole-g eligibility, g∘f values, f∘g values, first identity?, second identity?). If any g cell is untyped, use not-evaluated in the last four fields.

View answer and explanation

([1, 0, 0, 0, 0], [Yes, Yes, Yes], Yes, [2, 0, 0, 0, 0], [1, 0, 0], No, No)

Check all candidate cells, including unused ones, before evaluating either composition. A one-sided identity does not establish the other. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 0, 0, 0], [Yes, Yes, Yes], Yes, [2, 0, 0, 0, 0], [1, 0, 0], No, No).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-candidate-two-sided · candidate

35. Whole Candidate Inverse Type Check and Two Independent Identities

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 0, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Candidate g:Fin(3)→Fin(5) has g=[0, 2, 5]. Give (images, per-cell g type mask, whole-g eligibility, g∘f values, f∘g values, first identity?, second identity?). If any g cell is untyped, use not-evaluated in the last four fields.

View answer and explanation

([2, 0, 0, 0, 0], [Yes, Yes, No], No, not-evaluated, not-evaluated, not-evaluated, not-evaluated)

Check all candidate cells, including unused ones, before evaluating either composition. A one-sided identity does not establish the other. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 0, 0, 0], [Yes, Yes, No], No, not-evaluated, not-evaluated, not-evaluated, not-evaluated).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-candidate-two-sided · candidate

36. Whole Candidate Inverse Type Check and Two Independent Identities

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 1, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Candidate g:Fin(3)→Fin(5) has g=[0, 2, 4]. Give (images, per-cell g type mask, whole-g eligibility, g∘f values, f∘g values, first identity?, second identity?). If any g cell is untyped, use not-evaluated in the last four fields.

View answer and explanation

([0, 1, 0, 0, 0], [Yes, Yes, Yes], Yes, [0, 2, 0, 0, 0], [0, 0, 0], No, No)

Check all candidate cells, including unused ones, before evaluating either composition. A one-sided identity does not establish the other. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 0, 0, 0], [Yes, Yes, Yes], Yes, [0, 2, 0, 0, 0], [0, 0, 0], No, No).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-candidate-two-sided · candidate

37. Whole Candidate Inverse Type Check and Two Independent Identities

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 1, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Candidate g:Fin(3)→Fin(5) has g=[0, 2, 5]. Give (images, per-cell g type mask, whole-g eligibility, g∘f values, f∘g values, first identity?, second identity?). If any g cell is untyped, use not-evaluated in the last four fields.

View answer and explanation

([1, 1, 0, 0, 0], [Yes, Yes, No], No, not-evaluated, not-evaluated, not-evaluated, not-evaluated)

Check all candidate cells, including unused ones, before evaluating either composition. A one-sided identity does not establish the other. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 0, 0, 0], [Yes, Yes, No], No, not-evaluated, not-evaluated, not-evaluated, not-evaluated).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-candidate-two-sided · candidate

38. Whole Candidate Inverse Type Check and Two Independent Identities

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 1, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Candidate g:Fin(3)→Fin(5) has g=[0, 2, 4]. Give (images, per-cell g type mask, whole-g eligibility, g∘f values, f∘g values, first identity?, second identity?). If any g cell is untyped, use not-evaluated in the last four fields.

View answer and explanation

([2, 1, 0, 0, 0], [Yes, Yes, Yes], Yes, [4, 2, 0, 0, 0], [2, 0, 0], No, No)

Check all candidate cells, including unused ones, before evaluating either composition. A one-sided identity does not establish the other. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 0, 0, 0], [Yes, Yes, Yes], Yes, [4, 2, 0, 0, 0], [2, 0, 0], No, No).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-candidate-two-sided · candidate

39. Whole Candidate Inverse Type Check and Two Independent Identities

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 2, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Candidate g:Fin(3)→Fin(5) has g=[0, 2, 5]. Give (images, per-cell g type mask, whole-g eligibility, g∘f values, f∘g values, first identity?, second identity?). If any g cell is untyped, use not-evaluated in the last four fields.

View answer and explanation

([0, 2, 0, 0, 0], [Yes, Yes, No], No, not-evaluated, not-evaluated, not-evaluated, not-evaluated)

Check all candidate cells, including unused ones, before evaluating either composition. A one-sided identity does not establish the other. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 0, 0, 0], [Yes, Yes, No], No, not-evaluated, not-evaluated, not-evaluated, not-evaluated).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-candidate-two-sided · candidate

40. Whole Candidate Inverse Type Check and Two Independent Identities

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 2, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Candidate g:Fin(3)→Fin(5) has g=[0, 2, 4]. Give (images, per-cell g type mask, whole-g eligibility, g∘f values, f∘g values, first identity?, second identity?). If any g cell is untyped, use not-evaluated in the last four fields.

View answer and explanation

([1, 2, 0, 0, 0], [Yes, Yes, Yes], Yes, [2, 4, 0, 0, 0], [1, 0, 0], No, No)

Check all candidate cells, including unused ones, before evaluating either composition. A one-sided identity does not establish the other. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 0, 0, 0], [Yes, Yes, Yes], Yes, [2, 4, 0, 0, 0], [1, 0, 0], No, No).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-candidate-two-sided · candidate

41. Whole Candidate Inverse Type Check and Two Independent Identities

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 2, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Candidate g:Fin(3)→Fin(5) has g=[0, 2, 5]. Give (images, per-cell g type mask, whole-g eligibility, g∘f values, f∘g values, first identity?, second identity?). If any g cell is untyped, use not-evaluated in the last four fields.

View answer and explanation

([2, 2, 0, 0, 0], [Yes, Yes, No], No, not-evaluated, not-evaluated, not-evaluated, not-evaluated)

Check all candidate cells, including unused ones, before evaluating either composition. A one-sided identity does not establish the other. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2, 0, 0, 0], [Yes, Yes, No], No, not-evaluated, not-evaluated, not-evaluated, not-evaluated).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-candidate-two-sided · candidate

42. Whole Candidate Inverse Type Check and Two Independent Identities

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 0, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Candidate g:Fin(3)→Fin(5) has g=[0, 2, 4]. Give (images, per-cell g type mask, whole-g eligibility, g∘f values, f∘g values, first identity?, second identity?). If any g cell is untyped, use not-evaluated in the last four fields.

View answer and explanation

([0, 0, 1, 0, 0], [Yes, Yes, Yes], Yes, [0, 0, 2, 0, 0], [0, 1, 0], No, No)

Check all candidate cells, including unused ones, before evaluating either composition. A one-sided identity does not establish the other. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1, 0, 0], [Yes, Yes, Yes], Yes, [0, 0, 2, 0, 0], [0, 1, 0], No, No).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-candidate-two-sided · candidate

43. Whole Candidate Inverse Type Check and Two Independent Identities

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 0, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Candidate g:Fin(3)→Fin(5) has g=[0, 2, 5]. Give (images, per-cell g type mask, whole-g eligibility, g∘f values, f∘g values, first identity?, second identity?). If any g cell is untyped, use not-evaluated in the last four fields.

View answer and explanation

([1, 0, 1, 0, 0], [Yes, Yes, No], No, not-evaluated, not-evaluated, not-evaluated, not-evaluated)

Check all candidate cells, including unused ones, before evaluating either composition. A one-sided identity does not establish the other. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 1, 0, 0], [Yes, Yes, No], No, not-evaluated, not-evaluated, not-evaluated, not-evaluated).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-candidate-two-sided · candidate

44. Whole Candidate Inverse Type Check and Two Independent Identities

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 0, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Candidate g:Fin(3)→Fin(5) has g=[0, 2, 4]. Give (images, per-cell g type mask, whole-g eligibility, g∘f values, f∘g values, first identity?, second identity?). If any g cell is untyped, use not-evaluated in the last four fields.

View answer and explanation

([2, 0, 1, 0, 0], [Yes, Yes, Yes], Yes, [4, 0, 2, 0, 0], [2, 1, 0], No, No)

Check all candidate cells, including unused ones, before evaluating either composition. A one-sided identity does not establish the other. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 1, 0, 0], [Yes, Yes, Yes], Yes, [4, 0, 2, 0, 0], [2, 1, 0], No, No).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-candidate-two-sided · candidate

45. Whole Candidate Inverse Type Check and Two Independent Identities

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 1, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Candidate g:Fin(3)→Fin(5) has g=[0, 2, 5]. Give (images, per-cell g type mask, whole-g eligibility, g∘f values, f∘g values, first identity?, second identity?). If any g cell is untyped, use not-evaluated in the last four fields.

View answer and explanation

([0, 1, 1, 0, 0], [Yes, Yes, No], No, not-evaluated, not-evaluated, not-evaluated, not-evaluated)

Check all candidate cells, including unused ones, before evaluating either composition. A one-sided identity does not establish the other. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 1, 0, 0], [Yes, Yes, No], No, not-evaluated, not-evaluated, not-evaluated, not-evaluated).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-candidate-two-sided · candidate

46. Whole Candidate Inverse Type Check and Two Independent Identities

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 1, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Candidate g:Fin(3)→Fin(5) has g=[0, 2, 4]. Give (images, per-cell g type mask, whole-g eligibility, g∘f values, f∘g values, first identity?, second identity?). If any g cell is untyped, use not-evaluated in the last four fields.

View answer and explanation

([1, 1, 1, 0, 0], [Yes, Yes, Yes], Yes, [2, 2, 2, 0, 0], [1, 1, 0], No, No)

Check all candidate cells, including unused ones, before evaluating either composition. A one-sided identity does not establish the other. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 1, 0, 0], [Yes, Yes, Yes], Yes, [2, 2, 2, 0, 0], [1, 1, 0], No, No).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-candidate-two-sided · candidate

47. Whole Candidate Inverse Type Check and Two Independent Identities

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 1, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Candidate g:Fin(3)→Fin(5) has g=[0, 2, 5]. Give (images, per-cell g type mask, whole-g eligibility, g∘f values, f∘g values, first identity?, second identity?). If any g cell is untyped, use not-evaluated in the last four fields.

View answer and explanation

([2, 1, 1, 0, 0], [Yes, Yes, No], No, not-evaluated, not-evaluated, not-evaluated, not-evaluated)

Check all candidate cells, including unused ones, before evaluating either composition. A one-sided identity does not establish the other. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 1, 0, 0], [Yes, Yes, No], No, not-evaluated, not-evaluated, not-evaluated, not-evaluated).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-candidate-two-sided · candidate

48. Whole Candidate Inverse Type Check and Two Independent Identities

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 2, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Candidate g:Fin(3)→Fin(5) has g=[0, 2, 4]. Give (images, per-cell g type mask, whole-g eligibility, g∘f values, f∘g values, first identity?, second identity?). If any g cell is untyped, use not-evaluated in the last four fields.

View answer and explanation

([0, 2, 1, 0, 0], [Yes, Yes, Yes], Yes, [0, 4, 2, 0, 0], [0, 1, 0], No, No)

Check all candidate cells, including unused ones, before evaluating either composition. A one-sided identity does not establish the other. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 1, 0, 0], [Yes, Yes, Yes], Yes, [0, 4, 2, 0, 0], [0, 1, 0], No, No).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-candidate-two-sided · candidate

49. Whole Candidate Inverse Type Check and Two Independent Identities

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 2, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Candidate g:Fin(3)→Fin(5) has g=[0, 2, 5]. Give (images, per-cell g type mask, whole-g eligibility, g∘f values, f∘g values, first identity?, second identity?). If any g cell is untyped, use not-evaluated in the last four fields.

View answer and explanation

([1, 2, 1, 0, 0], [Yes, Yes, No], No, not-evaluated, not-evaluated, not-evaluated, not-evaluated)

Check all candidate cells, including unused ones, before evaluating either composition. A one-sided identity does not establish the other. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 1, 0, 0], [Yes, Yes, No], No, not-evaluated, not-evaluated, not-evaluated, not-evaluated).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-candidate-two-sided · candidate

50. Whole Candidate Inverse Type Check and Two Independent Identities

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 2, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Candidate g:Fin(3)→Fin(5) has g=[0, 2, 4]. Give (images, per-cell g type mask, whole-g eligibility, g∘f values, f∘g values, first identity?, second identity?). If any g cell is untyped, use not-evaluated in the last four fields.

View answer and explanation

([2, 2, 1, 0, 0], [Yes, Yes, Yes], Yes, [4, 4, 2, 0, 0], [2, 1, 0], No, No)

Check all candidate cells, including unused ones, before evaluating either composition. A one-sided identity does not establish the other. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2, 1, 0, 0], [Yes, Yes, Yes], Yes, [4, 4, 2, 0, 0], [2, 1, 0], No, No).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-candidate-two-sided · candidate

51. Inverse Existence and Two Codomain Query Guards

Use the complete total map Fin(0)=∅→Fin(0) with images=[] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Ordered queries=[0, -1]. Give (images, bijectivity, two codomain type guards, two inverse values). If f is not bijective use not-an-inverse-function for each value; otherwise an untyped query yields outside-codomain.

View answer and explanation

([], Yes, [No, No], [outside-codomain, outside-codomain])

Preserve the inverse-existence guard separately from the query masks; an inverse is not a reciprocal. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], Yes, [No, No], [outside-codomain, outside-codomain]).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-query-qualification · candidate

52. Inverse Existence and Two Codomain Query Guards

Use the complete total map Fin(0)=∅→Fin(2) with images=[] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Ordered queries=[0, 1]. Give (images, bijectivity, two codomain type guards, two inverse values). If f is not bijective use not-an-inverse-function for each value; otherwise an untyped query yields outside-codomain.

View answer and explanation

([], No, [Yes, Yes], [not-an-inverse-function, not-an-inverse-function])

Preserve the inverse-existence guard separately from the query masks; an inverse is not a reciprocal. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], No, [Yes, Yes], [not-an-inverse-function, not-an-inverse-function]).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-query-qualification · candidate

53. Inverse Existence and Two Codomain Query Guards

Use the complete total map Fin(1)={0,…,0}→Fin(1) with images=[0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Ordered queries=[0, 1]. Give (images, bijectivity, two codomain type guards, two inverse values). If f is not bijective use not-an-inverse-function for each value; otherwise an untyped query yields outside-codomain.

View answer and explanation

([0], Yes, [Yes, No], [0, outside-codomain])

Preserve the inverse-existence guard separately from the query masks; an inverse is not a reciprocal. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0], Yes, [Yes, No], [0, outside-codomain]).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-query-qualification · candidate

54. Inverse Existence and Two Codomain Query Guards

Use the complete total map Fin(3)={0,…,2}→Fin(3) with images=[0, 1, 2] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Ordered queries=[2, 1]. Give (images, bijectivity, two codomain type guards, two inverse values). If f is not bijective use not-an-inverse-function for each value; otherwise an untyped query yields outside-codomain.

View answer and explanation

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

Preserve the inverse-existence guard separately from the query masks; an inverse is not a reciprocal. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2], Yes, [Yes, Yes], [2, 1]).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-query-qualification · candidate

55. Inverse Existence and Two Codomain Query Guards

Use the complete total map Fin(3)={0,…,2}→Fin(3) with images=[0, 0, 1] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Ordered queries=[2, -1]. Give (images, bijectivity, two codomain type guards, two inverse values). If f is not bijective use not-an-inverse-function for each value; otherwise an untyped query yields outside-codomain.

View answer and explanation

([0, 0, 1], No, [Yes, No], [not-an-inverse-function, not-an-inverse-function])

Preserve the inverse-existence guard separately from the query masks; an inverse is not a reciprocal. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1], No, [Yes, No], [not-an-inverse-function, not-an-inverse-function]).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-query-qualification · candidate

56. Inverse Existence and Two Codomain Query Guards

Use the complete total map Fin(3)={0,…,2}→Fin(2) with images=[0, 1, 1] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Ordered queries=[0, 2]. Give (images, bijectivity, two codomain type guards, two inverse values). If f is not bijective use not-an-inverse-function for each value; otherwise an untyped query yields outside-codomain.

View answer and explanation

([0, 1, 1], No, [Yes, No], [not-an-inverse-function, not-an-inverse-function])

Preserve the inverse-existence guard separately from the query masks; an inverse is not a reciprocal. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 1], No, [Yes, No], [not-an-inverse-function, not-an-inverse-function]).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-query-qualification · candidate

57. Inverse Existence and Two Codomain Query Guards

Use the complete total map Fin(2)={0,…,1}→Fin(3) with images=[0, 1] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Ordered queries=[0, 2]. Give (images, bijectivity, two codomain type guards, two inverse values). If f is not bijective use not-an-inverse-function for each value; otherwise an untyped query yields outside-codomain.

View answer and explanation

([0, 1], No, [Yes, Yes], [not-an-inverse-function, not-an-inverse-function])

Preserve the inverse-existence guard separately from the query masks; an inverse is not a reciprocal. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1], No, [Yes, Yes], [not-an-inverse-function, not-an-inverse-function]).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-query-qualification · candidate

58. Inverse Existence and Two Codomain Query Guards

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 0, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Ordered queries=[0, 2]. Give (images, bijectivity, two codomain type guards, two inverse values). If f is not bijective use not-an-inverse-function for each value; otherwise an untyped query yields outside-codomain.

View answer and explanation

([0, 0, 0, 0, 0], No, [Yes, Yes], [not-an-inverse-function, not-an-inverse-function])

Preserve the inverse-existence guard separately from the query masks; an inverse is not a reciprocal. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 0, 0, 0], No, [Yes, Yes], [not-an-inverse-function, not-an-inverse-function]).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-query-qualification · candidate

59. Inverse Existence and Two Codomain Query Guards

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 0, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Ordered queries=[0, 2]. Give (images, bijectivity, two codomain type guards, two inverse values). If f is not bijective use not-an-inverse-function for each value; otherwise an untyped query yields outside-codomain.

View answer and explanation

([1, 0, 0, 0, 0], No, [Yes, Yes], [not-an-inverse-function, not-an-inverse-function])

Preserve the inverse-existence guard separately from the query masks; an inverse is not a reciprocal. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 0, 0, 0], No, [Yes, Yes], [not-an-inverse-function, not-an-inverse-function]).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-query-qualification · candidate

60. Inverse Existence and Two Codomain Query Guards

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 0, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Ordered queries=[0, 2]. Give (images, bijectivity, two codomain type guards, two inverse values). If f is not bijective use not-an-inverse-function for each value; otherwise an untyped query yields outside-codomain.

View answer and explanation

([2, 0, 0, 0, 0], No, [Yes, Yes], [not-an-inverse-function, not-an-inverse-function])

Preserve the inverse-existence guard separately from the query masks; an inverse is not a reciprocal. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 0, 0, 0], No, [Yes, Yes], [not-an-inverse-function, not-an-inverse-function]).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-query-qualification · candidate

61. Inverse Existence and Two Codomain Query Guards

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 1, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Ordered queries=[0, 2]. Give (images, bijectivity, two codomain type guards, two inverse values). If f is not bijective use not-an-inverse-function for each value; otherwise an untyped query yields outside-codomain.

View answer and explanation

([0, 1, 0, 0, 0], No, [Yes, Yes], [not-an-inverse-function, not-an-inverse-function])

Preserve the inverse-existence guard separately from the query masks; an inverse is not a reciprocal. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 0, 0, 0], No, [Yes, Yes], [not-an-inverse-function, not-an-inverse-function]).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-query-qualification · candidate

62. Inverse Existence and Two Codomain Query Guards

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 1, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Ordered queries=[0, 2]. Give (images, bijectivity, two codomain type guards, two inverse values). If f is not bijective use not-an-inverse-function for each value; otherwise an untyped query yields outside-codomain.

View answer and explanation

([1, 1, 0, 0, 0], No, [Yes, Yes], [not-an-inverse-function, not-an-inverse-function])

Preserve the inverse-existence guard separately from the query masks; an inverse is not a reciprocal. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 0, 0, 0], No, [Yes, Yes], [not-an-inverse-function, not-an-inverse-function]).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-query-qualification · candidate

63. Inverse Existence and Two Codomain Query Guards

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 1, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Ordered queries=[0, 2]. Give (images, bijectivity, two codomain type guards, two inverse values). If f is not bijective use not-an-inverse-function for each value; otherwise an untyped query yields outside-codomain.

View answer and explanation

([2, 1, 0, 0, 0], No, [Yes, Yes], [not-an-inverse-function, not-an-inverse-function])

Preserve the inverse-existence guard separately from the query masks; an inverse is not a reciprocal. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 0, 0, 0], No, [Yes, Yes], [not-an-inverse-function, not-an-inverse-function]).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-query-qualification · candidate

64. Inverse Existence and Two Codomain Query Guards

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 2, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Ordered queries=[0, 2]. Give (images, bijectivity, two codomain type guards, two inverse values). If f is not bijective use not-an-inverse-function for each value; otherwise an untyped query yields outside-codomain.

View answer and explanation

([0, 2, 0, 0, 0], No, [Yes, Yes], [not-an-inverse-function, not-an-inverse-function])

Preserve the inverse-existence guard separately from the query masks; an inverse is not a reciprocal. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 0, 0, 0], No, [Yes, Yes], [not-an-inverse-function, not-an-inverse-function]).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-query-qualification · candidate

65. Inverse Existence and Two Codomain Query Guards

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 2, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Ordered queries=[0, 2]. Give (images, bijectivity, two codomain type guards, two inverse values). If f is not bijective use not-an-inverse-function for each value; otherwise an untyped query yields outside-codomain.

View answer and explanation

([1, 2, 0, 0, 0], No, [Yes, Yes], [not-an-inverse-function, not-an-inverse-function])

Preserve the inverse-existence guard separately from the query masks; an inverse is not a reciprocal. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 0, 0, 0], No, [Yes, Yes], [not-an-inverse-function, not-an-inverse-function]).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-query-qualification · candidate

66. Inverse Existence and Two Codomain Query Guards

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 2, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Ordered queries=[0, 2]. Give (images, bijectivity, two codomain type guards, two inverse values). If f is not bijective use not-an-inverse-function for each value; otherwise an untyped query yields outside-codomain.

View answer and explanation

([2, 2, 0, 0, 0], No, [Yes, Yes], [not-an-inverse-function, not-an-inverse-function])

Preserve the inverse-existence guard separately from the query masks; an inverse is not a reciprocal. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2, 0, 0, 0], No, [Yes, Yes], [not-an-inverse-function, not-an-inverse-function]).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-query-qualification · candidate

67. Inverse Existence and Two Codomain Query Guards

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 0, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Ordered queries=[0, 2]. Give (images, bijectivity, two codomain type guards, two inverse values). If f is not bijective use not-an-inverse-function for each value; otherwise an untyped query yields outside-codomain.

View answer and explanation

([0, 0, 1, 0, 0], No, [Yes, Yes], [not-an-inverse-function, not-an-inverse-function])

Preserve the inverse-existence guard separately from the query masks; an inverse is not a reciprocal. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1, 0, 0], No, [Yes, Yes], [not-an-inverse-function, not-an-inverse-function]).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-query-qualification · candidate

68. Inverse Existence and Two Codomain Query Guards

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 0, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Ordered queries=[0, 2]. Give (images, bijectivity, two codomain type guards, two inverse values). If f is not bijective use not-an-inverse-function for each value; otherwise an untyped query yields outside-codomain.

View answer and explanation

([1, 0, 1, 0, 0], No, [Yes, Yes], [not-an-inverse-function, not-an-inverse-function])

Preserve the inverse-existence guard separately from the query masks; an inverse is not a reciprocal. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 1, 0, 0], No, [Yes, Yes], [not-an-inverse-function, not-an-inverse-function]).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-query-qualification · candidate

69. Inverse Existence and Two Codomain Query Guards

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 0, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Ordered queries=[0, 2]. Give (images, bijectivity, two codomain type guards, two inverse values). If f is not bijective use not-an-inverse-function for each value; otherwise an untyped query yields outside-codomain.

View answer and explanation

([2, 0, 1, 0, 0], No, [Yes, Yes], [not-an-inverse-function, not-an-inverse-function])

Preserve the inverse-existence guard separately from the query masks; an inverse is not a reciprocal. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 1, 0, 0], No, [Yes, Yes], [not-an-inverse-function, not-an-inverse-function]).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-query-qualification · candidate

70. Inverse Existence and Two Codomain Query Guards

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 1, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Ordered queries=[0, 2]. Give (images, bijectivity, two codomain type guards, two inverse values). If f is not bijective use not-an-inverse-function for each value; otherwise an untyped query yields outside-codomain.

View answer and explanation

([0, 1, 1, 0, 0], No, [Yes, Yes], [not-an-inverse-function, not-an-inverse-function])

Preserve the inverse-existence guard separately from the query masks; an inverse is not a reciprocal. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 1, 0, 0], No, [Yes, Yes], [not-an-inverse-function, not-an-inverse-function]).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-query-qualification · candidate

71. Inverse Existence and Two Codomain Query Guards

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 1, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Ordered queries=[0, 2]. Give (images, bijectivity, two codomain type guards, two inverse values). If f is not bijective use not-an-inverse-function for each value; otherwise an untyped query yields outside-codomain.

View answer and explanation

([1, 1, 1, 0, 0], No, [Yes, Yes], [not-an-inverse-function, not-an-inverse-function])

Preserve the inverse-existence guard separately from the query masks; an inverse is not a reciprocal. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 1, 0, 0], No, [Yes, Yes], [not-an-inverse-function, not-an-inverse-function]).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-query-qualification · candidate

72. Inverse Existence and Two Codomain Query Guards

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 1, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Ordered queries=[0, 2]. Give (images, bijectivity, two codomain type guards, two inverse values). If f is not bijective use not-an-inverse-function for each value; otherwise an untyped query yields outside-codomain.

View answer and explanation

([2, 1, 1, 0, 0], No, [Yes, Yes], [not-an-inverse-function, not-an-inverse-function])

Preserve the inverse-existence guard separately from the query masks; an inverse is not a reciprocal. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 1, 0, 0], No, [Yes, Yes], [not-an-inverse-function, not-an-inverse-function]).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-query-qualification · candidate

73. Inverse Existence and Two Codomain Query Guards

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 2, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Ordered queries=[0, 2]. Give (images, bijectivity, two codomain type guards, two inverse values). If f is not bijective use not-an-inverse-function for each value; otherwise an untyped query yields outside-codomain.

View answer and explanation

([0, 2, 1, 0, 0], No, [Yes, Yes], [not-an-inverse-function, not-an-inverse-function])

Preserve the inverse-existence guard separately from the query masks; an inverse is not a reciprocal. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 1, 0, 0], No, [Yes, Yes], [not-an-inverse-function, not-an-inverse-function]).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-query-qualification · candidate

74. Inverse Existence and Two Codomain Query Guards

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 2, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Ordered queries=[0, 2]. Give (images, bijectivity, two codomain type guards, two inverse values). If f is not bijective use not-an-inverse-function for each value; otherwise an untyped query yields outside-codomain.

View answer and explanation

([1, 2, 1, 0, 0], No, [Yes, Yes], [not-an-inverse-function, not-an-inverse-function])

Preserve the inverse-existence guard separately from the query masks; an inverse is not a reciprocal. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 1, 0, 0], No, [Yes, Yes], [not-an-inverse-function, not-an-inverse-function]).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-query-qualification · candidate

75. Inverse Existence and Two Codomain Query Guards

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 2, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Ordered queries=[0, 2]. Give (images, bijectivity, two codomain type guards, two inverse values). If f is not bijective use not-an-inverse-function for each value; otherwise an untyped query yields outside-codomain.

View answer and explanation

([2, 2, 1, 0, 0], No, [Yes, Yes], [not-an-inverse-function, not-an-inverse-function])

Preserve the inverse-existence guard separately from the query masks; an inverse is not a reciprocal. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2, 1, 0, 0], No, [Yes, Yes], [not-an-inverse-function, not-an-inverse-function]).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-query-qualification · candidate

76. Nonzero-Slope Real Affine Inverse and Two Rational Round Trips

Declare f:R→R by f(x)=a*x+b with a=-2≠0, b=1; ordered rational inputs y=[1/-2, -3/2] have nonzero original denominators. Give ([inverse slope, inverse constant], recovered x values, f(x) values, do both return their original y?). Use reduced positive-denominator fractions, including /1.

View answer and explanation

([-1/2, 1/2], [3/4, 5/4], [-1/2, -3/2], Yes)

Recover x=(y−b)/a and substitute into the original affine map. Negative original input denominators are normalized; inverse and reciprocal remain different operations. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1/2, 1/2], [3/4, 5/4], [-1/2, -3/2], Yes).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-affine-recovery · candidate

77. Nonzero-Slope Real Affine Inverse and Two Rational Round Trips

Declare f:R→R by f(x)=a*x+b with a=-2≠0, b=-11; ordered rational inputs y=[-11/3, 2/2] have nonzero original denominators. Give ([inverse slope, inverse constant], recovered x values, f(x) values, do both return their original y?). Use reduced positive-denominator fractions, including /1.

View answer and explanation

([-1/2, -11/2], [-11/3, -6/1], [-11/3, 1/1], Yes)

Recover x=(y−b)/a and substitute into the original affine map. Negative original input denominators are normalized; inverse and reciprocal remain different operations. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1/2, -11/2], [-11/3, -6/1], [-11/3, 1/1], Yes).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-affine-recovery · candidate

78. Nonzero-Slope Real Affine Inverse and Two Rational Round Trips

Declare f:R→R by f(x)=a*x+b with a=3≠0, b=-10; ordered rational inputs y=[-10/3, 3/3] have nonzero original denominators. Give ([inverse slope, inverse constant], recovered x values, f(x) values, do both return their original y?). Use reduced positive-denominator fractions, including /1.

View answer and explanation

([1/3, 10/3], [20/9, 11/3], [-10/3, 1/1], Yes)

Recover x=(y−b)/a and substitute into the original affine map. Negative original input denominators are normalized; inverse and reciprocal remain different operations. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1/3, 10/3], [20/9, 11/3], [-10/3, 1/1], Yes).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-affine-recovery · candidate

79. Nonzero-Slope Real Affine Inverse and Two Rational Round Trips

Declare f:R→R by f(x)=a*x+b with a=-4≠0, b=-9; ordered rational inputs y=[-9/3, 4/4] have nonzero original denominators. Give ([inverse slope, inverse constant], recovered x values, f(x) values, do both return their original y?). Use reduced positive-denominator fractions, including /1.

View answer and explanation

([-1/4, -9/4], [-3/2, -5/2], [-3/1, 1/1], Yes)

Recover x=(y−b)/a and substitute into the original affine map. Negative original input denominators are normalized; inverse and reciprocal remain different operations. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1/4, -9/4], [-3/2, -5/2], [-3/1, 1/1], Yes).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-affine-recovery · candidate

80. Nonzero-Slope Real Affine Inverse and Two Rational Round Trips

Declare f:R→R by f(x)=a*x+b with a=5≠0, b=-8; ordered rational inputs y=[-8/3, 5/1] have nonzero original denominators. Give ([inverse slope, inverse constant], recovered x values, f(x) values, do both return their original y?). Use reduced positive-denominator fractions, including /1.

View answer and explanation

([1/5, 8/5], [16/15, 13/5], [-8/3, 5/1], Yes)

Recover x=(y−b)/a and substitute into the original affine map. Negative original input denominators are normalized; inverse and reciprocal remain different operations. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1/5, 8/5], [16/15, 13/5], [-8/3, 5/1], Yes).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-affine-recovery · candidate

81. Nonzero-Slope Real Affine Inverse and Two Rational Round Trips

Declare f:R→R by f(x)=a*x+b with a=-1≠0, b=-7; ordered rational inputs y=[-7/3, 6/2] have nonzero original denominators. Give ([inverse slope, inverse constant], recovered x values, f(x) values, do both return their original y?). Use reduced positive-denominator fractions, including /1.

View answer and explanation

([-1/1, -7/1], [-14/3, -10/1], [-7/3, 3/1], Yes)

Recover x=(y−b)/a and substitute into the original affine map. Negative original input denominators are normalized; inverse and reciprocal remain different operations. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1/1, -7/1], [-14/3, -10/1], [-7/3, 3/1], Yes).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-affine-recovery · candidate

82. Nonzero-Slope Real Affine Inverse and Two Rational Round Trips

Declare f:R→R by f(x)=a*x+b with a=2≠0, b=-6; ordered rational inputs y=[-6/3, 7/3] have nonzero original denominators. Give ([inverse slope, inverse constant], recovered x values, f(x) values, do both return their original y?). Use reduced positive-denominator fractions, including /1.

View answer and explanation

([1/2, 3/1], [2/1, 25/6], [-2/1, 7/3], Yes)

Recover x=(y−b)/a and substitute into the original affine map. Negative original input denominators are normalized; inverse and reciprocal remain different operations. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1/2, 3/1], [2/1, 25/6], [-2/1, 7/3], Yes).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-affine-recovery · candidate

83. Nonzero-Slope Real Affine Inverse and Two Rational Round Trips

Declare f:R→R by f(x)=a*x+b with a=-3≠0, b=-5; ordered rational inputs y=[-5/3, 8/4] have nonzero original denominators. Give ([inverse slope, inverse constant], recovered x values, f(x) values, do both return their original y?). Use reduced positive-denominator fractions, including /1.

View answer and explanation

([-1/3, -5/3], [-10/9, -7/3], [-5/3, 2/1], Yes)

Recover x=(y−b)/a and substitute into the original affine map. Negative original input denominators are normalized; inverse and reciprocal remain different operations. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1/3, -5/3], [-10/9, -7/3], [-5/3, 2/1], Yes).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-affine-recovery · candidate

84. Nonzero-Slope Real Affine Inverse and Two Rational Round Trips

Declare f:R→R by f(x)=a*x+b with a=4≠0, b=-4; ordered rational inputs y=[-4/3, 9/1] have nonzero original denominators. Give ([inverse slope, inverse constant], recovered x values, f(x) values, do both return their original y?). Use reduced positive-denominator fractions, including /1.

View answer and explanation

([1/4, 1/1], [2/3, 13/4], [-4/3, 9/1], Yes)

Recover x=(y−b)/a and substitute into the original affine map. Negative original input denominators are normalized; inverse and reciprocal remain different operations. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1/4, 1/1], [2/3, 13/4], [-4/3, 9/1], Yes).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-affine-recovery · candidate

85. Nonzero-Slope Real Affine Inverse and Two Rational Round Trips

Declare f:R→R by f(x)=a*x+b with a=-5≠0, b=-3; ordered rational inputs y=[-3/3, 10/2] have nonzero original denominators. Give ([inverse slope, inverse constant], recovered x values, f(x) values, do both return their original y?). Use reduced positive-denominator fractions, including /1.

View answer and explanation

([-1/5, -3/5], [-2/5, -8/5], [-1/1, 5/1], Yes)

Recover x=(y−b)/a and substitute into the original affine map. Negative original input denominators are normalized; inverse and reciprocal remain different operations. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1/5, -3/5], [-2/5, -8/5], [-1/1, 5/1], Yes).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-affine-recovery · candidate

86. Nonzero-Slope Real Affine Inverse and Two Rational Round Trips

Declare f:R→R by f(x)=a*x+b with a=1≠0, b=-2; ordered rational inputs y=[-2/3, 11/3] have nonzero original denominators. Give ([inverse slope, inverse constant], recovered x values, f(x) values, do both return their original y?). Use reduced positive-denominator fractions, including /1.

View answer and explanation

([1/1, 2/1], [4/3, 17/3], [-2/3, 11/3], Yes)

Recover x=(y−b)/a and substitute into the original affine map. Negative original input denominators are normalized; inverse and reciprocal remain different operations. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1/1, 2/1], [4/3, 17/3], [-2/3, 11/3], Yes).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-affine-recovery · candidate

87. Nonzero-Slope Real Affine Inverse and Two Rational Round Trips

Declare f:R→R by f(x)=a*x+b with a=-2≠0, b=-1; ordered rational inputs y=[-1/3, 12/4] have nonzero original denominators. Give ([inverse slope, inverse constant], recovered x values, f(x) values, do both return their original y?). Use reduced positive-denominator fractions, including /1.

View answer and explanation

([-1/2, -1/2], [-1/3, -2/1], [-1/3, 3/1], Yes)

Recover x=(y−b)/a and substitute into the original affine map. Negative original input denominators are normalized; inverse and reciprocal remain different operations. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1/2, -1/2], [-1/3, -2/1], [-1/3, 3/1], Yes).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-affine-recovery · candidate

88. Nonzero-Slope Real Affine Inverse and Two Rational Round Trips

Declare f:R→R by f(x)=a*x+b with a=3≠0, b=0; ordered rational inputs y=[0/3, 13/1] have nonzero original denominators. Give ([inverse slope, inverse constant], recovered x values, f(x) values, do both return their original y?). Use reduced positive-denominator fractions, including /1.

View answer and explanation

([1/3, 0/1], [0/1, 13/3], [0/1, 13/1], Yes)

Recover x=(y−b)/a and substitute into the original affine map. Negative original input denominators are normalized; inverse and reciprocal remain different operations. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1/3, 0/1], [0/1, 13/3], [0/1, 13/1], Yes).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-affine-recovery · candidate

89. Nonzero-Slope Real Affine Inverse and Two Rational Round Trips

Declare f:R→R by f(x)=a*x+b with a=-4≠0, b=1; ordered rational inputs y=[1/3, 14/2] have nonzero original denominators. Give ([inverse slope, inverse constant], recovered x values, f(x) values, do both return their original y?). Use reduced positive-denominator fractions, including /1.

View answer and explanation

([-1/4, 1/4], [1/6, -3/2], [1/3, 7/1], Yes)

Recover x=(y−b)/a and substitute into the original affine map. Negative original input denominators are normalized; inverse and reciprocal remain different operations. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1/4, 1/4], [1/6, -3/2], [1/3, 7/1], Yes).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-affine-recovery · candidate

90. Nonzero-Slope Real Affine Inverse and Two Rational Round Trips

Declare f:R→R by f(x)=a*x+b with a=5≠0, b=2; ordered rational inputs y=[2/3, 15/3] have nonzero original denominators. Give ([inverse slope, inverse constant], recovered x values, f(x) values, do both return their original y?). Use reduced positive-denominator fractions, including /1.

View answer and explanation

([1/5, -2/5], [-4/15, 3/5], [2/3, 5/1], Yes)

Recover x=(y−b)/a and substitute into the original affine map. Negative original input denominators are normalized; inverse and reciprocal remain different operations. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1/5, -2/5], [-4/15, 3/5], [2/3, 5/1], Yes).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-affine-recovery · candidate

91. Nonzero-Slope Real Affine Inverse and Two Rational Round Trips

Declare f:R→R by f(x)=a*x+b with a=-1≠0, b=3; ordered rational inputs y=[3/3, 16/4] have nonzero original denominators. Give ([inverse slope, inverse constant], recovered x values, f(x) values, do both return their original y?). Use reduced positive-denominator fractions, including /1.

View answer and explanation

([-1/1, 3/1], [2/1, -1/1], [1/1, 4/1], Yes)

Recover x=(y−b)/a and substitute into the original affine map. Negative original input denominators are normalized; inverse and reciprocal remain different operations. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1/1, 3/1], [2/1, -1/1], [1/1, 4/1], Yes).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-affine-recovery · candidate

92. Nonzero-Slope Real Affine Inverse and Two Rational Round Trips

Declare f:R→R by f(x)=a*x+b with a=2≠0, b=4; ordered rational inputs y=[4/3, 17/1] have nonzero original denominators. Give ([inverse slope, inverse constant], recovered x values, f(x) values, do both return their original y?). Use reduced positive-denominator fractions, including /1.

View answer and explanation

([1/2, -2/1], [-4/3, 13/2], [4/3, 17/1], Yes)

Recover x=(y−b)/a and substitute into the original affine map. Negative original input denominators are normalized; inverse and reciprocal remain different operations. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1/2, -2/1], [-4/3, 13/2], [4/3, 17/1], Yes).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-affine-recovery · candidate

93. Nonzero-Slope Real Affine Inverse and Two Rational Round Trips

Declare f:R→R by f(x)=a*x+b with a=-3≠0, b=5; ordered rational inputs y=[5/3, 18/2] have nonzero original denominators. Give ([inverse slope, inverse constant], recovered x values, f(x) values, do both return their original y?). Use reduced positive-denominator fractions, including /1.

View answer and explanation

([-1/3, 5/3], [10/9, -4/3], [5/3, 9/1], Yes)

Recover x=(y−b)/a and substitute into the original affine map. Negative original input denominators are normalized; inverse and reciprocal remain different operations. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1/3, 5/3], [10/9, -4/3], [5/3, 9/1], Yes).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-affine-recovery · candidate

94. Nonzero-Slope Real Affine Inverse and Two Rational Round Trips

Declare f:R→R by f(x)=a*x+b with a=4≠0, b=6; ordered rational inputs y=[6/3, 19/3] have nonzero original denominators. Give ([inverse slope, inverse constant], recovered x values, f(x) values, do both return their original y?). Use reduced positive-denominator fractions, including /1.

View answer and explanation

([1/4, -3/2], [-1/1, 1/12], [2/1, 19/3], Yes)

Recover x=(y−b)/a and substitute into the original affine map. Negative original input denominators are normalized; inverse and reciprocal remain different operations. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1/4, -3/2], [-1/1, 1/12], [2/1, 19/3], Yes).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-affine-recovery · candidate

95. Nonzero-Slope Real Affine Inverse and Two Rational Round Trips

Declare f:R→R by f(x)=a*x+b with a=-5≠0, b=7; ordered rational inputs y=[7/3, 20/4] have nonzero original denominators. Give ([inverse slope, inverse constant], recovered x values, f(x) values, do both return their original y?). Use reduced positive-denominator fractions, including /1.

View answer and explanation

([-1/5, 7/5], [14/15, 2/5], [7/3, 5/1], Yes)

Recover x=(y−b)/a and substitute into the original affine map. Negative original input denominators are normalized; inverse and reciprocal remain different operations. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1/5, 7/5], [14/15, 2/5], [7/3, 5/1], Yes).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-affine-recovery · candidate

96. Nonzero-Slope Real Affine Inverse and Two Rational Round Trips

Declare f:R→R by f(x)=a*x+b with a=1≠0, b=8; ordered rational inputs y=[8/3, 21/1] have nonzero original denominators. Give ([inverse slope, inverse constant], recovered x values, f(x) values, do both return their original y?). Use reduced positive-denominator fractions, including /1.

View answer and explanation

([1/1, -8/1], [-16/3, 13/1], [8/3, 21/1], Yes)

Recover x=(y−b)/a and substitute into the original affine map. Negative original input denominators are normalized; inverse and reciprocal remain different operations. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1/1, -8/1], [-16/3, 13/1], [8/3, 21/1], Yes).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-affine-recovery · candidate

97. Nonzero-Slope Real Affine Inverse and Two Rational Round Trips

Declare f:R→R by f(x)=a*x+b with a=-2≠0, b=9; ordered rational inputs y=[9/3, 22/2] have nonzero original denominators. Give ([inverse slope, inverse constant], recovered x values, f(x) values, do both return their original y?). Use reduced positive-denominator fractions, including /1.

View answer and explanation

([-1/2, 9/2], [3/1, -1/1], [3/1, 11/1], Yes)

Recover x=(y−b)/a and substitute into the original affine map. Negative original input denominators are normalized; inverse and reciprocal remain different operations. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1/2, 9/2], [3/1, -1/1], [3/1, 11/1], Yes).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-affine-recovery · candidate

98. Nonzero-Slope Real Affine Inverse and Two Rational Round Trips

Declare f:R→R by f(x)=a*x+b with a=3≠0, b=10; ordered rational inputs y=[10/3, 23/3] have nonzero original denominators. Give ([inverse slope, inverse constant], recovered x values, f(x) values, do both return their original y?). Use reduced positive-denominator fractions, including /1.

View answer and explanation

([1/3, -10/3], [-20/9, -7/9], [10/3, 23/3], Yes)

Recover x=(y−b)/a and substitute into the original affine map. Negative original input denominators are normalized; inverse and reciprocal remain different operations. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1/3, -10/3], [-20/9, -7/9], [10/3, 23/3], Yes).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-affine-recovery · candidate

99. Nonzero-Slope Real Affine Inverse and Two Rational Round Trips

Declare f:R→R by f(x)=a*x+b with a=-4≠0, b=11; ordered rational inputs y=[11/3, 24/4] have nonzero original denominators. Give ([inverse slope, inverse constant], recovered x values, f(x) values, do both return their original y?). Use reduced positive-denominator fractions, including /1.

View answer and explanation

([-1/4, 11/4], [11/6, 5/4], [11/3, 6/1], Yes)

Recover x=(y−b)/a and substitute into the original affine map. Negative original input denominators are normalized; inverse and reciprocal remain different operations. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1/4, 11/4], [11/6, 5/4], [11/3, 6/1], Yes).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-affine-recovery · candidate

100. Nonzero-Slope Real Affine Inverse and Two Rational Round Trips

Declare f:R→R by f(x)=a*x+b with a=5≠0, b=12; ordered rational inputs y=[12/3, 25/1] have nonzero original denominators. Give ([inverse slope, inverse constant], recovered x values, f(x) values, do both return their original y?). Use reduced positive-denominator fractions, including /1.

View answer and explanation

([1/5, -12/5], [-8/5, 13/5], [4/1, 25/1], Yes)

Recover x=(y−b)/a and substitute into the original affine map. Negative original input denominators are normalized; inverse and reciprocal remain different operations. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1/5, -12/5], [-8/5, 13/5], [4/1, 25/1], Yes).

Reviewed step-by-step derivation

Inverse functions of bijections · qf-density17-inverse-affine-recovery · candidate

Canonical question JSONL · English JSONL · Validation report