# Complete Typed Function Data and Table Protocols

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-20261011-advance-20261015-r03-domain-codomain-range-foundations · 2026-10-11 · 100 questions

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



## 1. Complete Domain, Codomain, Graph, and Range Inventory

Fin(0)=∅ is the source and Fin(0)=∅ the codomain; complete typed images=[] are indexed from 0. All entries must be typed, even unused ones. Give ([n,m], all ordered [x,images[x]] rows, increasing range, unused codomain, range size).

View answer and explanation([0, 0], [], [], [], 0)

Read every source output; deduplicate outputs for the range and subtract from the codomain. A total mathematical table is not a partial program function. The exact result, retaining the requested order, is ([0, 0], [], [], [], 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0001/)

Domain, codomain, and range · qf-density15-typed-domain-range-inventory · candidate





## 2. Complete Domain, Codomain, Graph, and Range Inventory

Fin(3)={0,…,2} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[0, 0, 0] are indexed from 0. All entries must be typed, even unused ones. Give ([n,m], all ordered [x,images[x]] rows, increasing range, unused codomain, range size).

View answer and explanation([3, 3], [[0, 0], [1, 0], [2, 0]], [0], [1, 2], 1)

Read every source output; deduplicate outputs for the range and subtract from the codomain. A total mathematical table is not a partial program function. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [1, 0], [2, 0]], [0], [1, 2], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0002/)

Domain, codomain, and range · qf-density15-typed-domain-range-inventory · candidate





## 3. Complete Domain, Codomain, Graph, and Range Inventory

Fin(3)={0,…,2} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[1, 0, 0] are indexed from 0. All entries must be typed, even unused ones. Give ([n,m], all ordered [x,images[x]] rows, increasing range, unused codomain, range size).

View answer and explanation([3, 3], [[0, 1], [1, 0], [2, 0]], [0, 1], [2], 2)

Read every source output; deduplicate outputs for the range and subtract from the codomain. A total mathematical table is not a partial program function. The exact result, retaining the requested order, is ([3, 3], [[0, 1], [1, 0], [2, 0]], [0, 1], [2], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0003/)

Domain, codomain, and range · qf-density15-typed-domain-range-inventory · candidate





## 4. Complete Domain, Codomain, Graph, and Range Inventory

Fin(3)={0,…,2} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[2, 0, 0] are indexed from 0. All entries must be typed, even unused ones. Give ([n,m], all ordered [x,images[x]] rows, increasing range, unused codomain, range size).

View answer and explanation([3, 3], [[0, 2], [1, 0], [2, 0]], [0, 2], [1], 2)

Read every source output; deduplicate outputs for the range and subtract from the codomain. A total mathematical table is not a partial program function. The exact result, retaining the requested order, is ([3, 3], [[0, 2], [1, 0], [2, 0]], [0, 2], [1], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0004/)

Domain, codomain, and range · qf-density15-typed-domain-range-inventory · candidate





## 5. Complete Domain, Codomain, Graph, and Range Inventory

Fin(3)={0,…,2} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[0, 1, 0] are indexed from 0. All entries must be typed, even unused ones. Give ([n,m], all ordered [x,images[x]] rows, increasing range, unused codomain, range size).

View answer and explanation([3, 3], [[0, 0], [1, 1], [2, 0]], [0, 1], [2], 2)

Read every source output; deduplicate outputs for the range and subtract from the codomain. A total mathematical table is not a partial program function. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [1, 1], [2, 0]], [0, 1], [2], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0005/)

Domain, codomain, and range · qf-density15-typed-domain-range-inventory · candidate





## 6. Complete Domain, Codomain, Graph, and Range Inventory

Fin(3)={0,…,2} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[1, 1, 0] are indexed from 0. All entries must be typed, even unused ones. Give ([n,m], all ordered [x,images[x]] rows, increasing range, unused codomain, range size).

View answer and explanation([3, 3], [[0, 1], [1, 1], [2, 0]], [0, 1], [2], 2)

Read every source output; deduplicate outputs for the range and subtract from the codomain. A total mathematical table is not a partial program function. The exact result, retaining the requested order, is ([3, 3], [[0, 1], [1, 1], [2, 0]], [0, 1], [2], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0006/)

Domain, codomain, and range · qf-density15-typed-domain-range-inventory · candidate





## 7. Complete Domain, Codomain, Graph, and Range Inventory

Fin(3)={0,…,2} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[2, 1, 0] are indexed from 0. All entries must be typed, even unused ones. Give ([n,m], all ordered [x,images[x]] rows, increasing range, unused codomain, range size).

View answer and explanation([3, 3], [[0, 2], [1, 1], [2, 0]], [0, 1, 2], [], 3)

Read every source output; deduplicate outputs for the range and subtract from the codomain. A total mathematical table is not a partial program function. The exact result, retaining the requested order, is ([3, 3], [[0, 2], [1, 1], [2, 0]], [0, 1, 2], [], 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0007/)

Domain, codomain, and range · qf-density15-typed-domain-range-inventory · candidate





## 8. Complete Domain, Codomain, Graph, and Range Inventory

Fin(3)={0,…,2} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[0, 2, 0] are indexed from 0. All entries must be typed, even unused ones. Give ([n,m], all ordered [x,images[x]] rows, increasing range, unused codomain, range size).

View answer and explanation([3, 3], [[0, 0], [1, 2], [2, 0]], [0, 2], [1], 2)

Read every source output; deduplicate outputs for the range and subtract from the codomain. A total mathematical table is not a partial program function. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [1, 2], [2, 0]], [0, 2], [1], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0008/)

Domain, codomain, and range · qf-density15-typed-domain-range-inventory · candidate





## 9. Complete Domain, Codomain, Graph, and Range Inventory

Fin(3)={0,…,2} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[1, 2, 0] are indexed from 0. All entries must be typed, even unused ones. Give ([n,m], all ordered [x,images[x]] rows, increasing range, unused codomain, range size).

View answer and explanation([3, 3], [[0, 1], [1, 2], [2, 0]], [0, 1, 2], [], 3)

Read every source output; deduplicate outputs for the range and subtract from the codomain. A total mathematical table is not a partial program function. The exact result, retaining the requested order, is ([3, 3], [[0, 1], [1, 2], [2, 0]], [0, 1, 2], [], 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0009/)

Domain, codomain, and range · qf-density15-typed-domain-range-inventory · candidate





## 10. Complete Domain, Codomain, Graph, and Range Inventory

Fin(3)={0,…,2} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[2, 2, 0] are indexed from 0. All entries must be typed, even unused ones. Give ([n,m], all ordered [x,images[x]] rows, increasing range, unused codomain, range size).

View answer and explanation([3, 3], [[0, 2], [1, 2], [2, 0]], [0, 2], [1], 2)

Read every source output; deduplicate outputs for the range and subtract from the codomain. A total mathematical table is not a partial program function. The exact result, retaining the requested order, is ([3, 3], [[0, 2], [1, 2], [2, 0]], [0, 2], [1], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0010/)

Domain, codomain, and range · qf-density15-typed-domain-range-inventory · candidate





## 11. Complete Domain, Codomain, Graph, and Range Inventory

Fin(3)={0,…,2} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[0, 0, 1] are indexed from 0. All entries must be typed, even unused ones. Give ([n,m], all ordered [x,images[x]] rows, increasing range, unused codomain, range size).

View answer and explanation([3, 3], [[0, 0], [1, 0], [2, 1]], [0, 1], [2], 2)

Read every source output; deduplicate outputs for the range and subtract from the codomain. A total mathematical table is not a partial program function. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [1, 0], [2, 1]], [0, 1], [2], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0011/)

Domain, codomain, and range · qf-density15-typed-domain-range-inventory · candidate





## 12. Complete Domain, Codomain, Graph, and Range Inventory

Fin(3)={0,…,2} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[1, 0, 1] are indexed from 0. All entries must be typed, even unused ones. Give ([n,m], all ordered [x,images[x]] rows, increasing range, unused codomain, range size).

View answer and explanation([3, 3], [[0, 1], [1, 0], [2, 1]], [0, 1], [2], 2)

Read every source output; deduplicate outputs for the range and subtract from the codomain. A total mathematical table is not a partial program function. The exact result, retaining the requested order, is ([3, 3], [[0, 1], [1, 0], [2, 1]], [0, 1], [2], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0012/)

Domain, codomain, and range · qf-density15-typed-domain-range-inventory · candidate





## 13. Complete Domain, Codomain, Graph, and Range Inventory

Fin(3)={0,…,2} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[2, 0, 1] are indexed from 0. All entries must be typed, even unused ones. Give ([n,m], all ordered [x,images[x]] rows, increasing range, unused codomain, range size).

View answer and explanation([3, 3], [[0, 2], [1, 0], [2, 1]], [0, 1, 2], [], 3)

Read every source output; deduplicate outputs for the range and subtract from the codomain. A total mathematical table is not a partial program function. The exact result, retaining the requested order, is ([3, 3], [[0, 2], [1, 0], [2, 1]], [0, 1, 2], [], 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0013/)

Domain, codomain, and range · qf-density15-typed-domain-range-inventory · candidate





## 14. Complete Domain, Codomain, Graph, and Range Inventory

Fin(3)={0,…,2} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[0, 1, 1] are indexed from 0. All entries must be typed, even unused ones. Give ([n,m], all ordered [x,images[x]] rows, increasing range, unused codomain, range size).

View answer and explanation([3, 3], [[0, 0], [1, 1], [2, 1]], [0, 1], [2], 2)

Read every source output; deduplicate outputs for the range and subtract from the codomain. A total mathematical table is not a partial program function. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [1, 1], [2, 1]], [0, 1], [2], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0014/)

Domain, codomain, and range · qf-density15-typed-domain-range-inventory · candidate





## 15. Complete Domain, Codomain, Graph, and Range Inventory

Fin(3)={0,…,2} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[1, 1, 1] are indexed from 0. All entries must be typed, even unused ones. Give ([n,m], all ordered [x,images[x]] rows, increasing range, unused codomain, range size).

View answer and explanation([3, 3], [[0, 1], [1, 1], [2, 1]], [1], [0, 2], 1)

Read every source output; deduplicate outputs for the range and subtract from the codomain. A total mathematical table is not a partial program function. The exact result, retaining the requested order, is ([3, 3], [[0, 1], [1, 1], [2, 1]], [1], [0, 2], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0015/)

Domain, codomain, and range · qf-density15-typed-domain-range-inventory · candidate





## 16. Complete Domain, Codomain, Graph, and Range Inventory

Fin(3)={0,…,2} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[2, 1, 1] are indexed from 0. All entries must be typed, even unused ones. Give ([n,m], all ordered [x,images[x]] rows, increasing range, unused codomain, range size).

View answer and explanation([3, 3], [[0, 2], [1, 1], [2, 1]], [1, 2], [0], 2)

Read every source output; deduplicate outputs for the range and subtract from the codomain. A total mathematical table is not a partial program function. The exact result, retaining the requested order, is ([3, 3], [[0, 2], [1, 1], [2, 1]], [1, 2], [0], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0016/)

Domain, codomain, and range · qf-density15-typed-domain-range-inventory · candidate





## 17. Complete Domain, Codomain, Graph, and Range Inventory

Fin(3)={0,…,2} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[0, 2, 1] are indexed from 0. All entries must be typed, even unused ones. Give ([n,m], all ordered [x,images[x]] rows, increasing range, unused codomain, range size).

View answer and explanation([3, 3], [[0, 0], [1, 2], [2, 1]], [0, 1, 2], [], 3)

Read every source output; deduplicate outputs for the range and subtract from the codomain. A total mathematical table is not a partial program function. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [1, 2], [2, 1]], [0, 1, 2], [], 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0017/)

Domain, codomain, and range · qf-density15-typed-domain-range-inventory · candidate





## 18. Complete Domain, Codomain, Graph, and Range Inventory

Fin(3)={0,…,2} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[1, 2, 1] are indexed from 0. All entries must be typed, even unused ones. Give ([n,m], all ordered [x,images[x]] rows, increasing range, unused codomain, range size).

View answer and explanation([3, 3], [[0, 1], [1, 2], [2, 1]], [1, 2], [0], 2)

Read every source output; deduplicate outputs for the range and subtract from the codomain. A total mathematical table is not a partial program function. The exact result, retaining the requested order, is ([3, 3], [[0, 1], [1, 2], [2, 1]], [1, 2], [0], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0018/)

Domain, codomain, and range · qf-density15-typed-domain-range-inventory · candidate





## 19. Complete Domain, Codomain, Graph, and Range Inventory

Fin(3)={0,…,2} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[2, 2, 1] are indexed from 0. All entries must be typed, even unused ones. Give ([n,m], all ordered [x,images[x]] rows, increasing range, unused codomain, range size).

View answer and explanation([3, 3], [[0, 2], [1, 2], [2, 1]], [1, 2], [0], 2)

Read every source output; deduplicate outputs for the range and subtract from the codomain. A total mathematical table is not a partial program function. The exact result, retaining the requested order, is ([3, 3], [[0, 2], [1, 2], [2, 1]], [1, 2], [0], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0019/)

Domain, codomain, and range · qf-density15-typed-domain-range-inventory · candidate





## 20. Complete Domain, Codomain, Graph, and Range Inventory

Fin(3)={0,…,2} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[0, 0, 2] are indexed from 0. All entries must be typed, even unused ones. Give ([n,m], all ordered [x,images[x]] rows, increasing range, unused codomain, range size).

View answer and explanation([3, 3], [[0, 0], [1, 0], [2, 2]], [0, 2], [1], 2)

Read every source output; deduplicate outputs for the range and subtract from the codomain. A total mathematical table is not a partial program function. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [1, 0], [2, 2]], [0, 2], [1], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0020/)

Domain, codomain, and range · qf-density15-typed-domain-range-inventory · candidate





## 21. Complete Domain, Codomain, Graph, and Range Inventory

Fin(3)={0,…,2} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[1, 0, 2] are indexed from 0. All entries must be typed, even unused ones. Give ([n,m], all ordered [x,images[x]] rows, increasing range, unused codomain, range size).

View answer and explanation([3, 3], [[0, 1], [1, 0], [2, 2]], [0, 1, 2], [], 3)

Read every source output; deduplicate outputs for the range and subtract from the codomain. A total mathematical table is not a partial program function. The exact result, retaining the requested order, is ([3, 3], [[0, 1], [1, 0], [2, 2]], [0, 1, 2], [], 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0021/)

Domain, codomain, and range · qf-density15-typed-domain-range-inventory · candidate





## 22. Complete Domain, Codomain, Graph, and Range Inventory

Fin(3)={0,…,2} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[2, 0, 2] are indexed from 0. All entries must be typed, even unused ones. Give ([n,m], all ordered [x,images[x]] rows, increasing range, unused codomain, range size).

View answer and explanation([3, 3], [[0, 2], [1, 0], [2, 2]], [0, 2], [1], 2)

Read every source output; deduplicate outputs for the range and subtract from the codomain. A total mathematical table is not a partial program function. The exact result, retaining the requested order, is ([3, 3], [[0, 2], [1, 0], [2, 2]], [0, 2], [1], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0022/)

Domain, codomain, and range · qf-density15-typed-domain-range-inventory · candidate





## 23. Complete Domain, Codomain, Graph, and Range Inventory

Fin(3)={0,…,2} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[0, 1, 2] are indexed from 0. All entries must be typed, even unused ones. Give ([n,m], all ordered [x,images[x]] rows, increasing range, unused codomain, range size).

View answer and explanation([3, 3], [[0, 0], [1, 1], [2, 2]], [0, 1, 2], [], 3)

Read every source output; deduplicate outputs for the range and subtract from the codomain. A total mathematical table is not a partial program function. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [1, 1], [2, 2]], [0, 1, 2], [], 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0023/)

Domain, codomain, and range · qf-density15-typed-domain-range-inventory · candidate





## 24. Complete Domain, Codomain, Graph, and Range Inventory

Fin(3)={0,…,2} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[1, 1, 2] are indexed from 0. All entries must be typed, even unused ones. Give ([n,m], all ordered [x,images[x]] rows, increasing range, unused codomain, range size).

View answer and explanation([3, 3], [[0, 1], [1, 1], [2, 2]], [1, 2], [0], 2)

Read every source output; deduplicate outputs for the range and subtract from the codomain. A total mathematical table is not a partial program function. The exact result, retaining the requested order, is ([3, 3], [[0, 1], [1, 1], [2, 2]], [1, 2], [0], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0024/)

Domain, codomain, and range · qf-density15-typed-domain-range-inventory · candidate





## 25. Complete Domain, Codomain, Graph, and Range Inventory

Fin(3)={0,…,2} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[2, 1, 2] are indexed from 0. All entries must be typed, even unused ones. Give ([n,m], all ordered [x,images[x]] rows, increasing range, unused codomain, range size).

View answer and explanation([3, 3], [[0, 2], [1, 1], [2, 2]], [1, 2], [0], 2)

Read every source output; deduplicate outputs for the range and subtract from the codomain. A total mathematical table is not a partial program function. The exact result, retaining the requested order, is ([3, 3], [[0, 2], [1, 1], [2, 2]], [1, 2], [0], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0025/)

Domain, codomain, and range · qf-density15-typed-domain-range-inventory · candidate





## 26. Two Typed Codomain Candidates and Their Preimages

Fin(0)=∅ is the source and Fin(0)=∅ the codomain; complete typed images=[] are indexed from 0. All entries must be typed, even unused ones. Ordered codomain candidates=[0, -1]. Give (candidate-type eligibility, preimage index lists or outside-codomain, sizes or outside-codomain). An outside value is not a legal empty preimage.

View answer and explanation([No, No], [outside-codomain, outside-codomain], [outside-codomain, outside-codomain])

Guard codomain typing before traversing every source output for equality; a legal unused codomain value has an empty preimage, while an outside candidate is ill-typed. The exact result, retaining the requested order, is ([No, No], [outside-codomain, outside-codomain], [outside-codomain, outside-codomain]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0026/)

Domain, codomain, and range · qf-density15-typed-mapping-preimage-two-values · candidate





## 27. Two Typed Codomain Candidates and Their Preimages

Fin(0)=∅ is the source and Fin(2)={0,…,1} the codomain; complete typed images=[] are indexed from 0. All entries must be typed, even unused ones. Ordered codomain candidates=[0, 1]. Give (candidate-type eligibility, preimage index lists or outside-codomain, sizes or outside-codomain). An outside value is not a legal empty preimage.

View answer and explanation([Yes, Yes], [[], []], [0, 0])

Guard codomain typing before traversing every source output for equality; a legal unused codomain value has an empty preimage, while an outside candidate is ill-typed. The exact result, retaining the requested order, is ([Yes, Yes], [[], []], [0, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0027/)

Domain, codomain, and range · qf-density15-typed-mapping-preimage-two-values · candidate





## 28. Two Typed Codomain Candidates and Their Preimages

Fin(3)={0,…,2} is the source and Fin(2)={0,…,1} the codomain; complete typed images=[0, 1, 0] are indexed from 0. All entries must be typed, even unused ones. Ordered codomain candidates=[2, 0]. Give (candidate-type eligibility, preimage index lists or outside-codomain, sizes or outside-codomain). An outside value is not a legal empty preimage.

View answer and explanation([No, Yes], [outside-codomain, [0, 2]], [outside-codomain, 2])

Guard codomain typing before traversing every source output for equality; a legal unused codomain value has an empty preimage, while an outside candidate is ill-typed. The exact result, retaining the requested order, is ([No, Yes], [outside-codomain, [0, 2]], [outside-codomain, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0028/)

Domain, codomain, and range · qf-density15-typed-mapping-preimage-two-values · candidate





## 29. Two Typed Codomain Candidates and Their Preimages

Fin(5)={0,…,4} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[0, 0, 0, 0, 0] are indexed from 0. All entries must be typed, even unused ones. Ordered codomain candidates=[0, 1]. Give (candidate-type eligibility, preimage index lists or outside-codomain, sizes or outside-codomain). An outside value is not a legal empty preimage.

View answer and explanation([Yes, Yes], [[0, 1, 2, 3, 4], []], [5, 0])

Guard codomain typing before traversing every source output for equality; a legal unused codomain value has an empty preimage, while an outside candidate is ill-typed. The exact result, retaining the requested order, is ([Yes, Yes], [[0, 1, 2, 3, 4], []], [5, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0029/)

Domain, codomain, and range · qf-density15-typed-mapping-preimage-two-values · candidate





## 30. Two Typed Codomain Candidates and Their Preimages

Fin(5)={0,…,4} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[1, 0, 0, 0, 0] are indexed from 0. All entries must be typed, even unused ones. Ordered codomain candidates=[0, 1]. Give (candidate-type eligibility, preimage index lists or outside-codomain, sizes or outside-codomain). An outside value is not a legal empty preimage.

View answer and explanation([Yes, Yes], [[1, 2, 3, 4], [0]], [4, 1])

Guard codomain typing before traversing every source output for equality; a legal unused codomain value has an empty preimage, while an outside candidate is ill-typed. The exact result, retaining the requested order, is ([Yes, Yes], [[1, 2, 3, 4], [0]], [4, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0030/)

Domain, codomain, and range · qf-density15-typed-mapping-preimage-two-values · candidate





## 31. Two Typed Codomain Candidates and Their Preimages

Fin(5)={0,…,4} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[0, 1, 0, 0, 0] are indexed from 0. All entries must be typed, even unused ones. Ordered codomain candidates=[0, 1]. Give (candidate-type eligibility, preimage index lists or outside-codomain, sizes or outside-codomain). An outside value is not a legal empty preimage.

View answer and explanation([Yes, Yes], [[0, 2, 3, 4], [1]], [4, 1])

Guard codomain typing before traversing every source output for equality; a legal unused codomain value has an empty preimage, while an outside candidate is ill-typed. The exact result, retaining the requested order, is ([Yes, Yes], [[0, 2, 3, 4], [1]], [4, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0031/)

Domain, codomain, and range · qf-density15-typed-mapping-preimage-two-values · candidate





## 32. Two Typed Codomain Candidates and Their Preimages

Fin(5)={0,…,4} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[1, 1, 0, 0, 0] are indexed from 0. All entries must be typed, even unused ones. Ordered codomain candidates=[0, 1]. Give (candidate-type eligibility, preimage index lists or outside-codomain, sizes or outside-codomain). An outside value is not a legal empty preimage.

View answer and explanation([Yes, Yes], [[2, 3, 4], [0, 1]], [3, 2])

Guard codomain typing before traversing every source output for equality; a legal unused codomain value has an empty preimage, while an outside candidate is ill-typed. The exact result, retaining the requested order, is ([Yes, Yes], [[2, 3, 4], [0, 1]], [3, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0032/)

Domain, codomain, and range · qf-density15-typed-mapping-preimage-two-values · candidate





## 33. Two Typed Codomain Candidates and Their Preimages

Fin(5)={0,…,4} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[0, 0, 1, 0, 0] are indexed from 0. All entries must be typed, even unused ones. Ordered codomain candidates=[0, 1]. Give (candidate-type eligibility, preimage index lists or outside-codomain, sizes or outside-codomain). An outside value is not a legal empty preimage.

View answer and explanation([Yes, Yes], [[0, 1, 3, 4], [2]], [4, 1])

Guard codomain typing before traversing every source output for equality; a legal unused codomain value has an empty preimage, while an outside candidate is ill-typed. The exact result, retaining the requested order, is ([Yes, Yes], [[0, 1, 3, 4], [2]], [4, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0033/)

Domain, codomain, and range · qf-density15-typed-mapping-preimage-two-values · candidate





## 34. Two Typed Codomain Candidates and Their Preimages

Fin(5)={0,…,4} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[1, 0, 1, 0, 0] are indexed from 0. All entries must be typed, even unused ones. Ordered codomain candidates=[0, 1]. Give (candidate-type eligibility, preimage index lists or outside-codomain, sizes or outside-codomain). An outside value is not a legal empty preimage.

View answer and explanation([Yes, Yes], [[1, 3, 4], [0, 2]], [3, 2])

Guard codomain typing before traversing every source output for equality; a legal unused codomain value has an empty preimage, while an outside candidate is ill-typed. The exact result, retaining the requested order, is ([Yes, Yes], [[1, 3, 4], [0, 2]], [3, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0034/)

Domain, codomain, and range · qf-density15-typed-mapping-preimage-two-values · candidate





## 35. Two Typed Codomain Candidates and Their Preimages

Fin(5)={0,…,4} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[0, 1, 1, 0, 0] are indexed from 0. All entries must be typed, even unused ones. Ordered codomain candidates=[0, 1]. Give (candidate-type eligibility, preimage index lists or outside-codomain, sizes or outside-codomain). An outside value is not a legal empty preimage.

View answer and explanation([Yes, Yes], [[0, 3, 4], [1, 2]], [3, 2])

Guard codomain typing before traversing every source output for equality; a legal unused codomain value has an empty preimage, while an outside candidate is ill-typed. The exact result, retaining the requested order, is ([Yes, Yes], [[0, 3, 4], [1, 2]], [3, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0035/)

Domain, codomain, and range · qf-density15-typed-mapping-preimage-two-values · candidate





## 36. Two Typed Codomain Candidates and Their Preimages

Fin(5)={0,…,4} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[1, 1, 1, 0, 0] are indexed from 0. All entries must be typed, even unused ones. Ordered codomain candidates=[0, 1]. Give (candidate-type eligibility, preimage index lists or outside-codomain, sizes or outside-codomain). An outside value is not a legal empty preimage.

View answer and explanation([Yes, Yes], [[3, 4], [0, 1, 2]], [2, 3])

Guard codomain typing before traversing every source output for equality; a legal unused codomain value has an empty preimage, while an outside candidate is ill-typed. The exact result, retaining the requested order, is ([Yes, Yes], [[3, 4], [0, 1, 2]], [2, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0036/)

Domain, codomain, and range · qf-density15-typed-mapping-preimage-two-values · candidate





## 37. Two Typed Codomain Candidates and Their Preimages

Fin(5)={0,…,4} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[0, 0, 0, 1, 0] are indexed from 0. All entries must be typed, even unused ones. Ordered codomain candidates=[0, 1]. Give (candidate-type eligibility, preimage index lists or outside-codomain, sizes or outside-codomain). An outside value is not a legal empty preimage.

View answer and explanation([Yes, Yes], [[0, 1, 2, 4], [3]], [4, 1])

Guard codomain typing before traversing every source output for equality; a legal unused codomain value has an empty preimage, while an outside candidate is ill-typed. The exact result, retaining the requested order, is ([Yes, Yes], [[0, 1, 2, 4], [3]], [4, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0037/)

Domain, codomain, and range · qf-density15-typed-mapping-preimage-two-values · candidate





## 38. Two Typed Codomain Candidates and Their Preimages

Fin(5)={0,…,4} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[1, 0, 0, 1, 0] are indexed from 0. All entries must be typed, even unused ones. Ordered codomain candidates=[0, 1]. Give (candidate-type eligibility, preimage index lists or outside-codomain, sizes or outside-codomain). An outside value is not a legal empty preimage.

View answer and explanation([Yes, Yes], [[1, 2, 4], [0, 3]], [3, 2])

Guard codomain typing before traversing every source output for equality; a legal unused codomain value has an empty preimage, while an outside candidate is ill-typed. The exact result, retaining the requested order, is ([Yes, Yes], [[1, 2, 4], [0, 3]], [3, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0038/)

Domain, codomain, and range · qf-density15-typed-mapping-preimage-two-values · candidate





## 39. Two Typed Codomain Candidates and Their Preimages

Fin(5)={0,…,4} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[0, 1, 0, 1, 0] are indexed from 0. All entries must be typed, even unused ones. Ordered codomain candidates=[0, 1]. Give (candidate-type eligibility, preimage index lists or outside-codomain, sizes or outside-codomain). An outside value is not a legal empty preimage.

View answer and explanation([Yes, Yes], [[0, 2, 4], [1, 3]], [3, 2])

Guard codomain typing before traversing every source output for equality; a legal unused codomain value has an empty preimage, while an outside candidate is ill-typed. The exact result, retaining the requested order, is ([Yes, Yes], [[0, 2, 4], [1, 3]], [3, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0039/)

Domain, codomain, and range · qf-density15-typed-mapping-preimage-two-values · candidate





## 40. Two Typed Codomain Candidates and Their Preimages

Fin(5)={0,…,4} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[1, 1, 0, 1, 0] are indexed from 0. All entries must be typed, even unused ones. Ordered codomain candidates=[0, 1]. Give (candidate-type eligibility, preimage index lists or outside-codomain, sizes or outside-codomain). An outside value is not a legal empty preimage.

View answer and explanation([Yes, Yes], [[2, 4], [0, 1, 3]], [2, 3])

Guard codomain typing before traversing every source output for equality; a legal unused codomain value has an empty preimage, while an outside candidate is ill-typed. The exact result, retaining the requested order, is ([Yes, Yes], [[2, 4], [0, 1, 3]], [2, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0040/)

Domain, codomain, and range · qf-density15-typed-mapping-preimage-two-values · candidate





## 41. Two Typed Codomain Candidates and Their Preimages

Fin(5)={0,…,4} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[0, 0, 1, 1, 0] are indexed from 0. All entries must be typed, even unused ones. Ordered codomain candidates=[0, 1]. Give (candidate-type eligibility, preimage index lists or outside-codomain, sizes or outside-codomain). An outside value is not a legal empty preimage.

View answer and explanation([Yes, Yes], [[0, 1, 4], [2, 3]], [3, 2])

Guard codomain typing before traversing every source output for equality; a legal unused codomain value has an empty preimage, while an outside candidate is ill-typed. The exact result, retaining the requested order, is ([Yes, Yes], [[0, 1, 4], [2, 3]], [3, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0041/)

Domain, codomain, and range · qf-density15-typed-mapping-preimage-two-values · candidate





## 42. Two Typed Codomain Candidates and Their Preimages

Fin(5)={0,…,4} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[1, 0, 1, 1, 0] are indexed from 0. All entries must be typed, even unused ones. Ordered codomain candidates=[0, 1]. Give (candidate-type eligibility, preimage index lists or outside-codomain, sizes or outside-codomain). An outside value is not a legal empty preimage.

View answer and explanation([Yes, Yes], [[1, 4], [0, 2, 3]], [2, 3])

Guard codomain typing before traversing every source output for equality; a legal unused codomain value has an empty preimage, while an outside candidate is ill-typed. The exact result, retaining the requested order, is ([Yes, Yes], [[1, 4], [0, 2, 3]], [2, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0042/)

Domain, codomain, and range · qf-density15-typed-mapping-preimage-two-values · candidate





## 43. Two Typed Codomain Candidates and Their Preimages

Fin(5)={0,…,4} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[0, 1, 1, 1, 0] are indexed from 0. All entries must be typed, even unused ones. Ordered codomain candidates=[0, 1]. Give (candidate-type eligibility, preimage index lists or outside-codomain, sizes or outside-codomain). An outside value is not a legal empty preimage.

View answer and explanation([Yes, Yes], [[0, 4], [1, 2, 3]], [2, 3])

Guard codomain typing before traversing every source output for equality; a legal unused codomain value has an empty preimage, while an outside candidate is ill-typed. The exact result, retaining the requested order, is ([Yes, Yes], [[0, 4], [1, 2, 3]], [2, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0043/)

Domain, codomain, and range · qf-density15-typed-mapping-preimage-two-values · candidate





## 44. Two Typed Codomain Candidates and Their Preimages

Fin(5)={0,…,4} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[1, 1, 1, 1, 0] are indexed from 0. All entries must be typed, even unused ones. Ordered codomain candidates=[0, 1]. Give (candidate-type eligibility, preimage index lists or outside-codomain, sizes or outside-codomain). An outside value is not a legal empty preimage.

View answer and explanation([Yes, Yes], [[4], [0, 1, 2, 3]], [1, 4])

Guard codomain typing before traversing every source output for equality; a legal unused codomain value has an empty preimage, while an outside candidate is ill-typed. The exact result, retaining the requested order, is ([Yes, Yes], [[4], [0, 1, 2, 3]], [1, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0044/)

Domain, codomain, and range · qf-density15-typed-mapping-preimage-two-values · candidate





## 45. Two Typed Codomain Candidates and Their Preimages

Fin(5)={0,…,4} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[0, 0, 0, 0, 1] are indexed from 0. All entries must be typed, even unused ones. Ordered codomain candidates=[0, 1]. Give (candidate-type eligibility, preimage index lists or outside-codomain, sizes or outside-codomain). An outside value is not a legal empty preimage.

View answer and explanation([Yes, Yes], [[0, 1, 2, 3], [4]], [4, 1])

Guard codomain typing before traversing every source output for equality; a legal unused codomain value has an empty preimage, while an outside candidate is ill-typed. The exact result, retaining the requested order, is ([Yes, Yes], [[0, 1, 2, 3], [4]], [4, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0045/)

Domain, codomain, and range · qf-density15-typed-mapping-preimage-two-values · candidate





## 46. Two Typed Codomain Candidates and Their Preimages

Fin(5)={0,…,4} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[1, 0, 0, 0, 1] are indexed from 0. All entries must be typed, even unused ones. Ordered codomain candidates=[0, 1]. Give (candidate-type eligibility, preimage index lists or outside-codomain, sizes or outside-codomain). An outside value is not a legal empty preimage.

View answer and explanation([Yes, Yes], [[1, 2, 3], [0, 4]], [3, 2])

Guard codomain typing before traversing every source output for equality; a legal unused codomain value has an empty preimage, while an outside candidate is ill-typed. The exact result, retaining the requested order, is ([Yes, Yes], [[1, 2, 3], [0, 4]], [3, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0046/)

Domain, codomain, and range · qf-density15-typed-mapping-preimage-two-values · candidate





## 47. Two Typed Codomain Candidates and Their Preimages

Fin(5)={0,…,4} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[0, 1, 0, 0, 1] are indexed from 0. All entries must be typed, even unused ones. Ordered codomain candidates=[0, 1]. Give (candidate-type eligibility, preimage index lists or outside-codomain, sizes or outside-codomain). An outside value is not a legal empty preimage.

View answer and explanation([Yes, Yes], [[0, 2, 3], [1, 4]], [3, 2])

Guard codomain typing before traversing every source output for equality; a legal unused codomain value has an empty preimage, while an outside candidate is ill-typed. The exact result, retaining the requested order, is ([Yes, Yes], [[0, 2, 3], [1, 4]], [3, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0047/)

Domain, codomain, and range · qf-density15-typed-mapping-preimage-two-values · candidate





## 48. Two Typed Codomain Candidates and Their Preimages

Fin(5)={0,…,4} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[1, 1, 0, 0, 1] are indexed from 0. All entries must be typed, even unused ones. Ordered codomain candidates=[0, 1]. Give (candidate-type eligibility, preimage index lists or outside-codomain, sizes or outside-codomain). An outside value is not a legal empty preimage.

View answer and explanation([Yes, Yes], [[2, 3], [0, 1, 4]], [2, 3])

Guard codomain typing before traversing every source output for equality; a legal unused codomain value has an empty preimage, while an outside candidate is ill-typed. The exact result, retaining the requested order, is ([Yes, Yes], [[2, 3], [0, 1, 4]], [2, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0048/)

Domain, codomain, and range · qf-density15-typed-mapping-preimage-two-values · candidate





## 49. Two Typed Codomain Candidates and Their Preimages

Fin(5)={0,…,4} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[0, 0, 1, 0, 1] are indexed from 0. All entries must be typed, even unused ones. Ordered codomain candidates=[0, 1]. Give (candidate-type eligibility, preimage index lists or outside-codomain, sizes or outside-codomain). An outside value is not a legal empty preimage.

View answer and explanation([Yes, Yes], [[0, 1, 3], [2, 4]], [3, 2])

Guard codomain typing before traversing every source output for equality; a legal unused codomain value has an empty preimage, while an outside candidate is ill-typed. The exact result, retaining the requested order, is ([Yes, Yes], [[0, 1, 3], [2, 4]], [3, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0049/)

Domain, codomain, and range · qf-density15-typed-mapping-preimage-two-values · candidate





## 50. Two Typed Codomain Candidates and Their Preimages

Fin(5)={0,…,4} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[1, 0, 1, 0, 1] are indexed from 0. All entries must be typed, even unused ones. Ordered codomain candidates=[0, 1]. Give (candidate-type eligibility, preimage index lists or outside-codomain, sizes or outside-codomain). An outside value is not a legal empty preimage.

View answer and explanation([Yes, Yes], [[1, 3], [0, 2, 4]], [2, 3])

Guard codomain typing before traversing every source output for equality; a legal unused codomain value has an empty preimage, while an outside candidate is ill-typed. The exact result, retaining the requested order, is ([Yes, Yes], [[1, 3], [0, 2, 4]], [2, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0050/)

Domain, codomain, and range · qf-density15-typed-mapping-preimage-two-values · candidate





## 51. One Output Replacement and Range Additions/Removals

Fin(1)={0,…,0} is the source and Fin(1)={0,…,0} the codomain; complete typed images=[0] are indexed from 0. All entries must be typed, even unused ones. Replace at=0 by new value=0, after checking the whole original table. Give (old output, new table, new range, disappeared values, added values, m).

View answer and explanation(0, [0], [0], [], [], 1)

Retain other table entries and compute both directional range differences; replacing by the same value does not manufacture an addition. The exact result, retaining the requested order, is (0, [0], [0], [], [], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0051/)

Domain, codomain, and range · qf-density15-typed-mapping-output-replacement · candidate





## 52. One Output Replacement and Range Additions/Removals

Fin(2)={0,…,1} is the source and Fin(2)={0,…,1} the codomain; complete typed images=[0, 1] are indexed from 0. All entries must be typed, even unused ones. Replace at=0 by new value=1, after checking the whole original table. Give (old output, new table, new range, disappeared values, added values, m).

View answer and explanation(0, [1, 1], [1], [0], [], 2)

Retain other table entries and compute both directional range differences; replacing by the same value does not manufacture an addition. The exact result, retaining the requested order, is (0, [1, 1], [1], [0], [], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0052/)

Domain, codomain, and range · qf-density15-typed-mapping-output-replacement · candidate





## 53. One Output Replacement and Range Additions/Removals

Fin(2)={0,…,1} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[0, 0] are indexed from 0. All entries must be typed, even unused ones. Replace at=0 by new value=2, after checking the whole original table. Give (old output, new table, new range, disappeared values, added values, m).

View answer and explanation(0, [2, 0], [0, 2], [], [2], 3)

Retain other table entries and compute both directional range differences; replacing by the same value does not manufacture an addition. The exact result, retaining the requested order, is (0, [2, 0], [0, 2], [], [2], 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0053/)

Domain, codomain, and range · qf-density15-typed-mapping-output-replacement · candidate





## 54. One Output Replacement and Range Additions/Removals

Fin(2)={0,…,1} is the source and Fin(2)={0,…,1} the codomain; complete typed images=[0, 1] are indexed from 0. All entries must be typed, even unused ones. Replace at=1 by new value=1, after checking the whole original table. Give (old output, new table, new range, disappeared values, added values, m).

View answer and explanation(1, [0, 1], [0, 1], [], [], 2)

Retain other table entries and compute both directional range differences; replacing by the same value does not manufacture an addition. The exact result, retaining the requested order, is (1, [0, 1], [0, 1], [], [], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0054/)

Domain, codomain, and range · qf-density15-typed-mapping-output-replacement · candidate





## 55. One Output Replacement and Range Additions/Removals

Fin(3)={0,…,2} is the source and Fin(8)={0,…,7} the codomain; complete typed images=[4, 0, 5] are indexed from 0. All entries must be typed, even unused ones. Replace at=0 by new value=7, after checking the whole original table. Give (old output, new table, new range, disappeared values, added values, m).

View answer and explanation(4, [7, 0, 5], [0, 5, 7], [4], [7], 8)

Retain other table entries and compute both directional range differences; replacing by the same value does not manufacture an addition. The exact result, retaining the requested order, is (4, [7, 0, 5], [0, 5, 7], [4], [7], 8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0055/)

Domain, codomain, and range · qf-density15-typed-mapping-output-replacement · candidate





## 56. One Output Replacement and Range Additions/Removals

Fin(3)={0,…,2} is the source and Fin(8)={0,…,7} the codomain; complete typed images=[5, 0, 6] are indexed from 0. All entries must be typed, even unused ones. Replace at=0 by new value=0, after checking the whole original table. Give (old output, new table, new range, disappeared values, added values, m).

View answer and explanation(5, [0, 0, 6], [0, 6], [5], [], 8)

Retain other table entries and compute both directional range differences; replacing by the same value does not manufacture an addition. The exact result, retaining the requested order, is (5, [0, 0, 6], [0, 6], [5], [], 8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0056/)

Domain, codomain, and range · qf-density15-typed-mapping-output-replacement · candidate





## 57. One Output Replacement and Range Additions/Removals

Fin(3)={0,…,2} is the source and Fin(8)={0,…,7} the codomain; complete typed images=[6, 0, 7] are indexed from 0. All entries must be typed, even unused ones. Replace at=0 by new value=1, after checking the whole original table. Give (old output, new table, new range, disappeared values, added values, m).

View answer and explanation(6, [1, 0, 7], [0, 1, 7], [6], [1], 8)

Retain other table entries and compute both directional range differences; replacing by the same value does not manufacture an addition. The exact result, retaining the requested order, is (6, [1, 0, 7], [0, 1, 7], [6], [1], 8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0057/)

Domain, codomain, and range · qf-density15-typed-mapping-output-replacement · candidate





## 58. One Output Replacement and Range Additions/Removals

Fin(3)={0,…,2} is the source and Fin(8)={0,…,7} the codomain; complete typed images=[7, 0, 0] are indexed from 0. All entries must be typed, even unused ones. Replace at=0 by new value=2, after checking the whole original table. Give (old output, new table, new range, disappeared values, added values, m).

View answer and explanation(7, [2, 0, 0], [0, 2], [7], [2], 8)

Retain other table entries and compute both directional range differences; replacing by the same value does not manufacture an addition. The exact result, retaining the requested order, is (7, [2, 0, 0], [0, 2], [7], [2], 8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0058/)

Domain, codomain, and range · qf-density15-typed-mapping-output-replacement · candidate





## 59. One Output Replacement and Range Additions/Removals

Fin(3)={0,…,2} is the source and Fin(8)={0,…,7} the codomain; complete typed images=[0, 1, 1] are indexed from 0. All entries must be typed, even unused ones. Replace at=0 by new value=3, after checking the whole original table. Give (old output, new table, new range, disappeared values, added values, m).

View answer and explanation(0, [3, 1, 1], [1, 3], [0], [3], 8)

Retain other table entries and compute both directional range differences; replacing by the same value does not manufacture an addition. The exact result, retaining the requested order, is (0, [3, 1, 1], [1, 3], [0], [3], 8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0059/)

Domain, codomain, and range · qf-density15-typed-mapping-output-replacement · candidate





## 60. One Output Replacement and Range Additions/Removals

Fin(3)={0,…,2} is the source and Fin(8)={0,…,7} the codomain; complete typed images=[1, 1, 2] are indexed from 0. All entries must be typed, even unused ones. Replace at=0 by new value=4, after checking the whole original table. Give (old output, new table, new range, disappeared values, added values, m).

View answer and explanation(1, [4, 1, 2], [1, 2, 4], [], [4], 8)

Retain other table entries and compute both directional range differences; replacing by the same value does not manufacture an addition. The exact result, retaining the requested order, is (1, [4, 1, 2], [1, 2, 4], [], [4], 8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0060/)

Domain, codomain, and range · qf-density15-typed-mapping-output-replacement · candidate





## 61. One Output Replacement and Range Additions/Removals

Fin(3)={0,…,2} is the source and Fin(8)={0,…,7} the codomain; complete typed images=[2, 1, 3] are indexed from 0. All entries must be typed, even unused ones. Replace at=0 by new value=5, after checking the whole original table. Give (old output, new table, new range, disappeared values, added values, m).

View answer and explanation(2, [5, 1, 3], [1, 3, 5], [2], [5], 8)

Retain other table entries and compute both directional range differences; replacing by the same value does not manufacture an addition. The exact result, retaining the requested order, is (2, [5, 1, 3], [1, 3, 5], [2], [5], 8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0061/)

Domain, codomain, and range · qf-density15-typed-mapping-output-replacement · candidate





## 62. One Output Replacement and Range Additions/Removals

Fin(3)={0,…,2} is the source and Fin(8)={0,…,7} the codomain; complete typed images=[3, 1, 4] are indexed from 0. All entries must be typed, even unused ones. Replace at=0 by new value=6, after checking the whole original table. Give (old output, new table, new range, disappeared values, added values, m).

View answer and explanation(3, [6, 1, 4], [1, 4, 6], [3], [6], 8)

Retain other table entries and compute both directional range differences; replacing by the same value does not manufacture an addition. The exact result, retaining the requested order, is (3, [6, 1, 4], [1, 4, 6], [3], [6], 8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0062/)

Domain, codomain, and range · qf-density15-typed-mapping-output-replacement · candidate





## 63. One Output Replacement and Range Additions/Removals

Fin(3)={0,…,2} is the source and Fin(8)={0,…,7} the codomain; complete typed images=[4, 1, 5] are indexed from 0. All entries must be typed, even unused ones. Replace at=0 by new value=7, after checking the whole original table. Give (old output, new table, new range, disappeared values, added values, m).

View answer and explanation(4, [7, 1, 5], [1, 5, 7], [4], [7], 8)

Retain other table entries and compute both directional range differences; replacing by the same value does not manufacture an addition. The exact result, retaining the requested order, is (4, [7, 1, 5], [1, 5, 7], [4], [7], 8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0063/)

Domain, codomain, and range · qf-density15-typed-mapping-output-replacement · candidate





## 64. One Output Replacement and Range Additions/Removals

Fin(3)={0,…,2} is the source and Fin(8)={0,…,7} the codomain; complete typed images=[5, 1, 6] are indexed from 0. All entries must be typed, even unused ones. Replace at=0 by new value=0, after checking the whole original table. Give (old output, new table, new range, disappeared values, added values, m).

View answer and explanation(5, [0, 1, 6], [0, 1, 6], [5], [0], 8)

Retain other table entries and compute both directional range differences; replacing by the same value does not manufacture an addition. The exact result, retaining the requested order, is (5, [0, 1, 6], [0, 1, 6], [5], [0], 8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0064/)

Domain, codomain, and range · qf-density15-typed-mapping-output-replacement · candidate





## 65. One Output Replacement and Range Additions/Removals

Fin(3)={0,…,2} is the source and Fin(8)={0,…,7} the codomain; complete typed images=[6, 1, 7] are indexed from 0. All entries must be typed, even unused ones. Replace at=0 by new value=1, after checking the whole original table. Give (old output, new table, new range, disappeared values, added values, m).

View answer and explanation(6, [1, 1, 7], [1, 7], [6], [], 8)

Retain other table entries and compute both directional range differences; replacing by the same value does not manufacture an addition. The exact result, retaining the requested order, is (6, [1, 1, 7], [1, 7], [6], [], 8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0065/)

Domain, codomain, and range · qf-density15-typed-mapping-output-replacement · candidate





## 66. One Output Replacement and Range Additions/Removals

Fin(3)={0,…,2} is the source and Fin(8)={0,…,7} the codomain; complete typed images=[7, 1, 0] are indexed from 0. All entries must be typed, even unused ones. Replace at=0 by new value=2, after checking the whole original table. Give (old output, new table, new range, disappeared values, added values, m).

View answer and explanation(7, [2, 1, 0], [0, 1, 2], [7], [2], 8)

Retain other table entries and compute both directional range differences; replacing by the same value does not manufacture an addition. The exact result, retaining the requested order, is (7, [2, 1, 0], [0, 1, 2], [7], [2], 8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0066/)

Domain, codomain, and range · qf-density15-typed-mapping-output-replacement · candidate





## 67. One Output Replacement and Range Additions/Removals

Fin(3)={0,…,2} is the source and Fin(8)={0,…,7} the codomain; complete typed images=[0, 2, 1] are indexed from 0. All entries must be typed, even unused ones. Replace at=0 by new value=3, after checking the whole original table. Give (old output, new table, new range, disappeared values, added values, m).

View answer and explanation(0, [3, 2, 1], [1, 2, 3], [0], [3], 8)

Retain other table entries and compute both directional range differences; replacing by the same value does not manufacture an addition. The exact result, retaining the requested order, is (0, [3, 2, 1], [1, 2, 3], [0], [3], 8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0067/)

Domain, codomain, and range · qf-density15-typed-mapping-output-replacement · candidate





## 68. One Output Replacement and Range Additions/Removals

Fin(3)={0,…,2} is the source and Fin(8)={0,…,7} the codomain; complete typed images=[1, 2, 2] are indexed from 0. All entries must be typed, even unused ones. Replace at=0 by new value=4, after checking the whole original table. Give (old output, new table, new range, disappeared values, added values, m).

View answer and explanation(1, [4, 2, 2], [2, 4], [1], [4], 8)

Retain other table entries and compute both directional range differences; replacing by the same value does not manufacture an addition. The exact result, retaining the requested order, is (1, [4, 2, 2], [2, 4], [1], [4], 8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0068/)

Domain, codomain, and range · qf-density15-typed-mapping-output-replacement · candidate





## 69. One Output Replacement and Range Additions/Removals

Fin(3)={0,…,2} is the source and Fin(8)={0,…,7} the codomain; complete typed images=[2, 2, 3] are indexed from 0. All entries must be typed, even unused ones. Replace at=0 by new value=5, after checking the whole original table. Give (old output, new table, new range, disappeared values, added values, m).

View answer and explanation(2, [5, 2, 3], [2, 3, 5], [], [5], 8)

Retain other table entries and compute both directional range differences; replacing by the same value does not manufacture an addition. The exact result, retaining the requested order, is (2, [5, 2, 3], [2, 3, 5], [], [5], 8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0069/)

Domain, codomain, and range · qf-density15-typed-mapping-output-replacement · candidate





## 70. One Output Replacement and Range Additions/Removals

Fin(3)={0,…,2} is the source and Fin(8)={0,…,7} the codomain; complete typed images=[3, 2, 4] are indexed from 0. All entries must be typed, even unused ones. Replace at=0 by new value=6, after checking the whole original table. Give (old output, new table, new range, disappeared values, added values, m).

View answer and explanation(3, [6, 2, 4], [2, 4, 6], [3], [6], 8)

Retain other table entries and compute both directional range differences; replacing by the same value does not manufacture an addition. The exact result, retaining the requested order, is (3, [6, 2, 4], [2, 4, 6], [3], [6], 8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0070/)

Domain, codomain, and range · qf-density15-typed-mapping-output-replacement · candidate





## 71. One Output Replacement and Range Additions/Removals

Fin(3)={0,…,2} is the source and Fin(8)={0,…,7} the codomain; complete typed images=[4, 2, 5] are indexed from 0. All entries must be typed, even unused ones. Replace at=0 by new value=7, after checking the whole original table. Give (old output, new table, new range, disappeared values, added values, m).

View answer and explanation(4, [7, 2, 5], [2, 5, 7], [4], [7], 8)

Retain other table entries and compute both directional range differences; replacing by the same value does not manufacture an addition. The exact result, retaining the requested order, is (4, [7, 2, 5], [2, 5, 7], [4], [7], 8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0071/)

Domain, codomain, and range · qf-density15-typed-mapping-output-replacement · candidate





## 72. One Output Replacement and Range Additions/Removals

Fin(3)={0,…,2} is the source and Fin(8)={0,…,7} the codomain; complete typed images=[5, 2, 6] are indexed from 0. All entries must be typed, even unused ones. Replace at=0 by new value=0, after checking the whole original table. Give (old output, new table, new range, disappeared values, added values, m).

View answer and explanation(5, [0, 2, 6], [0, 2, 6], [5], [0], 8)

Retain other table entries and compute both directional range differences; replacing by the same value does not manufacture an addition. The exact result, retaining the requested order, is (5, [0, 2, 6], [0, 2, 6], [5], [0], 8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0072/)

Domain, codomain, and range · qf-density15-typed-mapping-output-replacement · candidate





## 73. One Output Replacement and Range Additions/Removals

Fin(3)={0,…,2} is the source and Fin(8)={0,…,7} the codomain; complete typed images=[6, 2, 7] are indexed from 0. All entries must be typed, even unused ones. Replace at=0 by new value=1, after checking the whole original table. Give (old output, new table, new range, disappeared values, added values, m).

View answer and explanation(6, [1, 2, 7], [1, 2, 7], [6], [1], 8)

Retain other table entries and compute both directional range differences; replacing by the same value does not manufacture an addition. The exact result, retaining the requested order, is (6, [1, 2, 7], [1, 2, 7], [6], [1], 8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0073/)

Domain, codomain, and range · qf-density15-typed-mapping-output-replacement · candidate





## 74. One Output Replacement and Range Additions/Removals

Fin(3)={0,…,2} is the source and Fin(8)={0,…,7} the codomain; complete typed images=[7, 2, 0] are indexed from 0. All entries must be typed, even unused ones. Replace at=0 by new value=2, after checking the whole original table. Give (old output, new table, new range, disappeared values, added values, m).

View answer and explanation(7, [2, 2, 0], [0, 2], [7], [], 8)

Retain other table entries and compute both directional range differences; replacing by the same value does not manufacture an addition. The exact result, retaining the requested order, is (7, [2, 2, 0], [0, 2], [7], [], 8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0074/)

Domain, codomain, and range · qf-density15-typed-mapping-output-replacement · candidate





## 75. One Output Replacement and Range Additions/Removals

Fin(3)={0,…,2} is the source and Fin(8)={0,…,7} the codomain; complete typed images=[0, 3, 1] are indexed from 0. All entries must be typed, even unused ones. Replace at=0 by new value=3, after checking the whole original table. Give (old output, new table, new range, disappeared values, added values, m).

View answer and explanation(0, [3, 3, 1], [1, 3], [0], [], 8)

Retain other table entries and compute both directional range differences; replacing by the same value does not manufacture an addition. The exact result, retaining the requested order, is (0, [3, 3, 1], [1, 3], [0], [], 8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0075/)

Domain, codomain, and range · qf-density15-typed-mapping-output-replacement · candidate





## 76. Candidate One-Row Protocol and Total-Table Eligibility

Source=Fin(0)=∅, codomain=Fin(0)=∅; each input must have exactly one data row. Candidate raw rows=[]. Give ([n,m], row-endpoint typing mask, raw occurrence counts for each legal input, missing inputs, repeated inputs, total-table eligibility, ordered images or not-total-table). Count by input before rejecting an outside output.

View answer and explanation([0, 0], [], [], [], [], Yes, [])

Audit every raw endpoint and each input’s row count separately; only a fully eligible representation becomes a total table. Equal-value duplicate rows violate this protocol but do not imply a multivalued mathematical function. The exact result, retaining the requested order, is ([0, 0], [], [], [], [], Yes, []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0076/)

Domain, codomain, and range · qf-density15-typed-mapping-table-candidate-audit · candidate





## 77. Candidate One-Row Protocol and Total-Table Eligibility

Source=Fin(1)={0,…,0}, codomain=Fin(1)={0,…,0}; each input must have exactly one data row. Candidate raw rows=[[0, 0]]. Give ([n,m], row-endpoint typing mask, raw occurrence counts for each legal input, missing inputs, repeated inputs, total-table eligibility, ordered images or not-total-table). Count by input before rejecting an outside output.

View answer and explanation([1, 1], [Yes], [1], [], [], Yes, [0])

Audit every raw endpoint and each input’s row count separately; only a fully eligible representation becomes a total table. Equal-value duplicate rows violate this protocol but do not imply a multivalued mathematical function. The exact result, retaining the requested order, is ([1, 1], [Yes], [1], [], [], Yes, [0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0077/)

Domain, codomain, and range · qf-density15-typed-mapping-table-candidate-audit · candidate





## 78. Candidate One-Row Protocol and Total-Table Eligibility

Source=Fin(1)={0,…,0}, codomain=Fin(1)={0,…,0}; each input must have exactly one data row. Candidate raw rows=[[0, 0], [0, 0]]. Give ([n,m], row-endpoint typing mask, raw occurrence counts for each legal input, missing inputs, repeated inputs, total-table eligibility, ordered images or not-total-table). Count by input before rejecting an outside output.

View answer and explanation([1, 1], [Yes, Yes], [2], [], [0], No, not-total-table)

Audit every raw endpoint and each input’s row count separately; only a fully eligible representation becomes a total table. Equal-value duplicate rows violate this protocol but do not imply a multivalued mathematical function. The exact result, retaining the requested order, is ([1, 1], [Yes, Yes], [2], [], [0], No, not-total-table).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0078/)

Domain, codomain, and range · qf-density15-typed-mapping-table-candidate-audit · candidate





## 79. Candidate One-Row Protocol and Total-Table Eligibility

Source=Fin(2)={0,…,1}, codomain=Fin(2)={0,…,1}; each input must have exactly one data row. Candidate raw rows=[[0, 0]]. Give ([n,m], row-endpoint typing mask, raw occurrence counts for each legal input, missing inputs, repeated inputs, total-table eligibility, ordered images or not-total-table). Count by input before rejecting an outside output.

View answer and explanation([2, 2], [Yes], [1, 0], [1], [], No, not-total-table)

Audit every raw endpoint and each input’s row count separately; only a fully eligible representation becomes a total table. Equal-value duplicate rows violate this protocol but do not imply a multivalued mathematical function. The exact result, retaining the requested order, is ([2, 2], [Yes], [1, 0], [1], [], No, not-total-table).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0079/)

Domain, codomain, and range · qf-density15-typed-mapping-table-candidate-audit · candidate





## 80. Candidate One-Row Protocol and Total-Table Eligibility

Source=Fin(2)={0,…,1}, codomain=Fin(2)={0,…,1}; each input must have exactly one data row. Candidate raw rows=[[0, 0], [1, 2]]. Give ([n,m], row-endpoint typing mask, raw occurrence counts for each legal input, missing inputs, repeated inputs, total-table eligibility, ordered images or not-total-table). Count by input before rejecting an outside output.

View answer and explanation([2, 2], [Yes, No], [1, 1], [], [], No, not-total-table)

Audit every raw endpoint and each input’s row count separately; only a fully eligible representation becomes a total table. Equal-value duplicate rows violate this protocol but do not imply a multivalued mathematical function. The exact result, retaining the requested order, is ([2, 2], [Yes, No], [1, 1], [], [], No, not-total-table).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0080/)

Domain, codomain, and range · qf-density15-typed-mapping-table-candidate-audit · candidate





## 81. Candidate One-Row Protocol and Total-Table Eligibility

Source=Fin(2)={0,…,1}, codomain=Fin(2)={0,…,1}; each input must have exactly one data row. Candidate raw rows=[[-1, 0], [0, 0], [1, 1]]. Give ([n,m], row-endpoint typing mask, raw occurrence counts for each legal input, missing inputs, repeated inputs, total-table eligibility, ordered images or not-total-table). Count by input before rejecting an outside output.

View answer and explanation([2, 2], [No, Yes, Yes], [1, 1], [], [], No, not-total-table)

Audit every raw endpoint and each input’s row count separately; only a fully eligible representation becomes a total table. Equal-value duplicate rows violate this protocol but do not imply a multivalued mathematical function. The exact result, retaining the requested order, is ([2, 2], [No, Yes, Yes], [1, 1], [], [], No, not-total-table).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0081/)

Domain, codomain, and range · qf-density15-typed-mapping-table-candidate-audit · candidate





## 82. Candidate One-Row Protocol and Total-Table Eligibility

Source=Fin(3)={0,…,2}, codomain=Fin(2)={0,…,1}; each input must have exactly one data row. Candidate raw rows=[]. Give ([n,m], row-endpoint typing mask, raw occurrence counts for each legal input, missing inputs, repeated inputs, total-table eligibility, ordered images or not-total-table). Count by input before rejecting an outside output.

View answer and explanation([3, 2], [], [0, 0, 0], [0, 1, 2], [], No, not-total-table)

Audit every raw endpoint and each input’s row count separately; only a fully eligible representation becomes a total table. Equal-value duplicate rows violate this protocol but do not imply a multivalued mathematical function. The exact result, retaining the requested order, is ([3, 2], [], [0, 0, 0], [0, 1, 2], [], No, not-total-table).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0082/)

Domain, codomain, and range · qf-density15-typed-mapping-table-candidate-audit · candidate





## 83. Candidate One-Row Protocol and Total-Table Eligibility

Source=Fin(3)={0,…,2}, codomain=Fin(2)={0,…,1}; each input must have exactly one data row. Candidate raw rows=[[0, 0]]. Give ([n,m], row-endpoint typing mask, raw occurrence counts for each legal input, missing inputs, repeated inputs, total-table eligibility, ordered images or not-total-table). Count by input before rejecting an outside output.

View answer and explanation([3, 2], [Yes], [1, 0, 0], [1, 2], [], No, not-total-table)

Audit every raw endpoint and each input’s row count separately; only a fully eligible representation becomes a total table. Equal-value duplicate rows violate this protocol but do not imply a multivalued mathematical function. The exact result, retaining the requested order, is ([3, 2], [Yes], [1, 0, 0], [1, 2], [], No, not-total-table).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0083/)

Domain, codomain, and range · qf-density15-typed-mapping-table-candidate-audit · candidate





## 84. Candidate One-Row Protocol and Total-Table Eligibility

Source=Fin(3)={0,…,2}, codomain=Fin(2)={0,…,1}; each input must have exactly one data row. Candidate raw rows=[[0, 0], [0, 0]]. Give ([n,m], row-endpoint typing mask, raw occurrence counts for each legal input, missing inputs, repeated inputs, total-table eligibility, ordered images or not-total-table). Count by input before rejecting an outside output.

View answer and explanation([3, 2], [Yes, Yes], [2, 0, 0], [1, 2], [0], No, not-total-table)

Audit every raw endpoint and each input’s row count separately; only a fully eligible representation becomes a total table. Equal-value duplicate rows violate this protocol but do not imply a multivalued mathematical function. The exact result, retaining the requested order, is ([3, 2], [Yes, Yes], [2, 0, 0], [1, 2], [0], No, not-total-table).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0084/)

Domain, codomain, and range · qf-density15-typed-mapping-table-candidate-audit · candidate





## 85. Candidate One-Row Protocol and Total-Table Eligibility

Source=Fin(3)={0,…,2}, codomain=Fin(2)={0,…,1}; each input must have exactly one data row. Candidate raw rows=[[1, 1]]. Give ([n,m], row-endpoint typing mask, raw occurrence counts for each legal input, missing inputs, repeated inputs, total-table eligibility, ordered images or not-total-table). Count by input before rejecting an outside output.

View answer and explanation([3, 2], [Yes], [0, 1, 0], [0, 2], [], No, not-total-table)

Audit every raw endpoint and each input’s row count separately; only a fully eligible representation becomes a total table. Equal-value duplicate rows violate this protocol but do not imply a multivalued mathematical function. The exact result, retaining the requested order, is ([3, 2], [Yes], [0, 1, 0], [0, 2], [], No, not-total-table).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0085/)

Domain, codomain, and range · qf-density15-typed-mapping-table-candidate-audit · candidate





## 86. Candidate One-Row Protocol and Total-Table Eligibility

Source=Fin(3)={0,…,2}, codomain=Fin(2)={0,…,1}; each input must have exactly one data row. Candidate raw rows=[[0, 0], [1, 1]]. Give ([n,m], row-endpoint typing mask, raw occurrence counts for each legal input, missing inputs, repeated inputs, total-table eligibility, ordered images or not-total-table). Count by input before rejecting an outside output.

View answer and explanation([3, 2], [Yes, Yes], [1, 1, 0], [2], [], No, not-total-table)

Audit every raw endpoint and each input’s row count separately; only a fully eligible representation becomes a total table. Equal-value duplicate rows violate this protocol but do not imply a multivalued mathematical function. The exact result, retaining the requested order, is ([3, 2], [Yes, Yes], [1, 1, 0], [2], [], No, not-total-table).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0086/)

Domain, codomain, and range · qf-density15-typed-mapping-table-candidate-audit · candidate





## 87. Candidate One-Row Protocol and Total-Table Eligibility

Source=Fin(3)={0,…,2}, codomain=Fin(2)={0,…,1}; each input must have exactly one data row. Candidate raw rows=[[0, 0], [0, 0], [1, 1]]. Give ([n,m], row-endpoint typing mask, raw occurrence counts for each legal input, missing inputs, repeated inputs, total-table eligibility, ordered images or not-total-table). Count by input before rejecting an outside output.

View answer and explanation([3, 2], [Yes, Yes, Yes], [2, 1, 0], [2], [0], No, not-total-table)

Audit every raw endpoint and each input’s row count separately; only a fully eligible representation becomes a total table. Equal-value duplicate rows violate this protocol but do not imply a multivalued mathematical function. The exact result, retaining the requested order, is ([3, 2], [Yes, Yes, Yes], [2, 1, 0], [2], [0], No, not-total-table).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0087/)

Domain, codomain, and range · qf-density15-typed-mapping-table-candidate-audit · candidate





## 88. Candidate One-Row Protocol and Total-Table Eligibility

Source=Fin(3)={0,…,2}, codomain=Fin(2)={0,…,1}; each input must have exactly one data row. Candidate raw rows=[[1, 1], [1, 1]]. Give ([n,m], row-endpoint typing mask, raw occurrence counts for each legal input, missing inputs, repeated inputs, total-table eligibility, ordered images or not-total-table). Count by input before rejecting an outside output.

View answer and explanation([3, 2], [Yes, Yes], [0, 2, 0], [0, 2], [1], No, not-total-table)

Audit every raw endpoint and each input’s row count separately; only a fully eligible representation becomes a total table. Equal-value duplicate rows violate this protocol but do not imply a multivalued mathematical function. The exact result, retaining the requested order, is ([3, 2], [Yes, Yes], [0, 2, 0], [0, 2], [1], No, not-total-table).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0088/)

Domain, codomain, and range · qf-density15-typed-mapping-table-candidate-audit · candidate





## 89. Candidate One-Row Protocol and Total-Table Eligibility

Source=Fin(3)={0,…,2}, codomain=Fin(2)={0,…,1}; each input must have exactly one data row. Candidate raw rows=[[0, 0], [1, 1], [1, 1]]. Give ([n,m], row-endpoint typing mask, raw occurrence counts for each legal input, missing inputs, repeated inputs, total-table eligibility, ordered images or not-total-table). Count by input before rejecting an outside output.

View answer and explanation([3, 2], [Yes, Yes, Yes], [1, 2, 0], [2], [1], No, not-total-table)

Audit every raw endpoint and each input’s row count separately; only a fully eligible representation becomes a total table. Equal-value duplicate rows violate this protocol but do not imply a multivalued mathematical function. The exact result, retaining the requested order, is ([3, 2], [Yes, Yes, Yes], [1, 2, 0], [2], [1], No, not-total-table).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0089/)

Domain, codomain, and range · qf-density15-typed-mapping-table-candidate-audit · candidate





## 90. Candidate One-Row Protocol and Total-Table Eligibility

Source=Fin(3)={0,…,2}, codomain=Fin(2)={0,…,1}; each input must have exactly one data row. Candidate raw rows=[[0, 0], [0, 0], [1, 1], [1, 1]]. Give ([n,m], row-endpoint typing mask, raw occurrence counts for each legal input, missing inputs, repeated inputs, total-table eligibility, ordered images or not-total-table). Count by input before rejecting an outside output.

View answer and explanation([3, 2], [Yes, Yes, Yes, Yes], [2, 2, 0], [2], [0, 1], No, not-total-table)

Audit every raw endpoint and each input’s row count separately; only a fully eligible representation becomes a total table. Equal-value duplicate rows violate this protocol but do not imply a multivalued mathematical function. The exact result, retaining the requested order, is ([3, 2], [Yes, Yes, Yes, Yes], [2, 2, 0], [2], [0, 1], No, not-total-table).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0090/)

Domain, codomain, and range · qf-density15-typed-mapping-table-candidate-audit · candidate





## 91. Candidate One-Row Protocol and Total-Table Eligibility

Source=Fin(3)={0,…,2}, codomain=Fin(2)={0,…,1}; each input must have exactly one data row. Candidate raw rows=[[2, 0]]. Give ([n,m], row-endpoint typing mask, raw occurrence counts for each legal input, missing inputs, repeated inputs, total-table eligibility, ordered images or not-total-table). Count by input before rejecting an outside output.

View answer and explanation([3, 2], [Yes], [0, 0, 1], [0, 1], [], No, not-total-table)

Audit every raw endpoint and each input’s row count separately; only a fully eligible representation becomes a total table. Equal-value duplicate rows violate this protocol but do not imply a multivalued mathematical function. The exact result, retaining the requested order, is ([3, 2], [Yes], [0, 0, 1], [0, 1], [], No, not-total-table).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0091/)

Domain, codomain, and range · qf-density15-typed-mapping-table-candidate-audit · candidate





## 92. Candidate One-Row Protocol and Total-Table Eligibility

Source=Fin(3)={0,…,2}, codomain=Fin(2)={0,…,1}; each input must have exactly one data row. Candidate raw rows=[[0, 0], [2, 0]]. Give ([n,m], row-endpoint typing mask, raw occurrence counts for each legal input, missing inputs, repeated inputs, total-table eligibility, ordered images or not-total-table). Count by input before rejecting an outside output.

View answer and explanation([3, 2], [Yes, Yes], [1, 0, 1], [1], [], No, not-total-table)

Audit every raw endpoint and each input’s row count separately; only a fully eligible representation becomes a total table. Equal-value duplicate rows violate this protocol but do not imply a multivalued mathematical function. The exact result, retaining the requested order, is ([3, 2], [Yes, Yes], [1, 0, 1], [1], [], No, not-total-table).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0092/)

Domain, codomain, and range · qf-density15-typed-mapping-table-candidate-audit · candidate





## 93. Candidate One-Row Protocol and Total-Table Eligibility

Source=Fin(3)={0,…,2}, codomain=Fin(2)={0,…,1}; each input must have exactly one data row. Candidate raw rows=[[0, 0], [0, 0], [2, 0]]. Give ([n,m], row-endpoint typing mask, raw occurrence counts for each legal input, missing inputs, repeated inputs, total-table eligibility, ordered images or not-total-table). Count by input before rejecting an outside output.

View answer and explanation([3, 2], [Yes, Yes, Yes], [2, 0, 1], [1], [0], No, not-total-table)

Audit every raw endpoint and each input’s row count separately; only a fully eligible representation becomes a total table. Equal-value duplicate rows violate this protocol but do not imply a multivalued mathematical function. The exact result, retaining the requested order, is ([3, 2], [Yes, Yes, Yes], [2, 0, 1], [1], [0], No, not-total-table).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0093/)

Domain, codomain, and range · qf-density15-typed-mapping-table-candidate-audit · candidate





## 94. Candidate One-Row Protocol and Total-Table Eligibility

Source=Fin(3)={0,…,2}, codomain=Fin(2)={0,…,1}; each input must have exactly one data row. Candidate raw rows=[[1, 1], [2, 0]]. Give ([n,m], row-endpoint typing mask, raw occurrence counts for each legal input, missing inputs, repeated inputs, total-table eligibility, ordered images or not-total-table). Count by input before rejecting an outside output.

View answer and explanation([3, 2], [Yes, Yes], [0, 1, 1], [0], [], No, not-total-table)

Audit every raw endpoint and each input’s row count separately; only a fully eligible representation becomes a total table. Equal-value duplicate rows violate this protocol but do not imply a multivalued mathematical function. The exact result, retaining the requested order, is ([3, 2], [Yes, Yes], [0, 1, 1], [0], [], No, not-total-table).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0094/)

Domain, codomain, and range · qf-density15-typed-mapping-table-candidate-audit · candidate





## 95. Candidate One-Row Protocol and Total-Table Eligibility

Source=Fin(3)={0,…,2}, codomain=Fin(2)={0,…,1}; each input must have exactly one data row. Candidate raw rows=[[0, 0], [1, 1], [2, 0]]. Give ([n,m], row-endpoint typing mask, raw occurrence counts for each legal input, missing inputs, repeated inputs, total-table eligibility, ordered images or not-total-table). Count by input before rejecting an outside output.

View answer and explanation([3, 2], [Yes, Yes, Yes], [1, 1, 1], [], [], Yes, [0, 1, 0])

Audit every raw endpoint and each input’s row count separately; only a fully eligible representation becomes a total table. Equal-value duplicate rows violate this protocol but do not imply a multivalued mathematical function. The exact result, retaining the requested order, is ([3, 2], [Yes, Yes, Yes], [1, 1, 1], [], [], Yes, [0, 1, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0095/)

Domain, codomain, and range · qf-density15-typed-mapping-table-candidate-audit · candidate





## 96. Candidate One-Row Protocol and Total-Table Eligibility

Source=Fin(3)={0,…,2}, codomain=Fin(2)={0,…,1}; each input must have exactly one data row. Candidate raw rows=[[0, 0], [0, 0], [1, 1], [2, 0]]. Give ([n,m], row-endpoint typing mask, raw occurrence counts for each legal input, missing inputs, repeated inputs, total-table eligibility, ordered images or not-total-table). Count by input before rejecting an outside output.

View answer and explanation([3, 2], [Yes, Yes, Yes, Yes], [2, 1, 1], [], [0], No, not-total-table)

Audit every raw endpoint and each input’s row count separately; only a fully eligible representation becomes a total table. Equal-value duplicate rows violate this protocol but do not imply a multivalued mathematical function. The exact result, retaining the requested order, is ([3, 2], [Yes, Yes, Yes, Yes], [2, 1, 1], [], [0], No, not-total-table).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0096/)

Domain, codomain, and range · qf-density15-typed-mapping-table-candidate-audit · candidate





## 97. Candidate One-Row Protocol and Total-Table Eligibility

Source=Fin(3)={0,…,2}, codomain=Fin(2)={0,…,1}; each input must have exactly one data row. Candidate raw rows=[[1, 1], [1, 1], [2, 0]]. Give ([n,m], row-endpoint typing mask, raw occurrence counts for each legal input, missing inputs, repeated inputs, total-table eligibility, ordered images or not-total-table). Count by input before rejecting an outside output.

View answer and explanation([3, 2], [Yes, Yes, Yes], [0, 2, 1], [0], [1], No, not-total-table)

Audit every raw endpoint and each input’s row count separately; only a fully eligible representation becomes a total table. Equal-value duplicate rows violate this protocol but do not imply a multivalued mathematical function. The exact result, retaining the requested order, is ([3, 2], [Yes, Yes, Yes], [0, 2, 1], [0], [1], No, not-total-table).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0097/)

Domain, codomain, and range · qf-density15-typed-mapping-table-candidate-audit · candidate





## 98. Candidate One-Row Protocol and Total-Table Eligibility

Source=Fin(3)={0,…,2}, codomain=Fin(2)={0,…,1}; each input must have exactly one data row. Candidate raw rows=[[0, 0], [1, 1], [1, 1], [2, 0]]. Give ([n,m], row-endpoint typing mask, raw occurrence counts for each legal input, missing inputs, repeated inputs, total-table eligibility, ordered images or not-total-table). Count by input before rejecting an outside output.

View answer and explanation([3, 2], [Yes, Yes, Yes, Yes], [1, 2, 1], [], [1], No, not-total-table)

Audit every raw endpoint and each input’s row count separately; only a fully eligible representation becomes a total table. Equal-value duplicate rows violate this protocol but do not imply a multivalued mathematical function. The exact result, retaining the requested order, is ([3, 2], [Yes, Yes, Yes, Yes], [1, 2, 1], [], [1], No, not-total-table).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0098/)

Domain, codomain, and range · qf-density15-typed-mapping-table-candidate-audit · candidate





## 99. Candidate One-Row Protocol and Total-Table Eligibility

Source=Fin(3)={0,…,2}, codomain=Fin(2)={0,…,1}; each input must have exactly one data row. Candidate raw rows=[[0, 0], [0, 0], [1, 1], [1, 1], [2, 0]]. Give ([n,m], row-endpoint typing mask, raw occurrence counts for each legal input, missing inputs, repeated inputs, total-table eligibility, ordered images or not-total-table). Count by input before rejecting an outside output.

View answer and explanation([3, 2], [Yes, Yes, Yes, Yes, Yes], [2, 2, 1], [], [0, 1], No, not-total-table)

Audit every raw endpoint and each input’s row count separately; only a fully eligible representation becomes a total table. Equal-value duplicate rows violate this protocol but do not imply a multivalued mathematical function. The exact result, retaining the requested order, is ([3, 2], [Yes, Yes, Yes, Yes, Yes], [2, 2, 1], [], [0, 1], No, not-total-table).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0099/)

Domain, codomain, and range · qf-density15-typed-mapping-table-candidate-audit · candidate





## 100. Candidate One-Row Protocol and Total-Table Eligibility

Source=Fin(3)={0,…,2}, codomain=Fin(2)={0,…,1}; each input must have exactly one data row. Candidate raw rows=[[2, 0], [2, 0]]. Give ([n,m], row-endpoint typing mask, raw occurrence counts for each legal input, missing inputs, repeated inputs, total-table eligibility, ordered images or not-total-table). Count by input before rejecting an outside output.

View answer and explanation([3, 2], [Yes, Yes], [0, 0, 2], [0, 1], [2], No, not-total-table)

Audit every raw endpoint and each input’s row count separately; only a fully eligible representation becomes a total table. Equal-value duplicate rows violate this protocol but do not imply a multivalued mathematical function. The exact result, retaining the requested order, is ([3, 2], [Yes, Yes], [0, 0, 2], [0, 1], [2], No, not-total-table).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261011-advance-20261015-r03-domain-codomain-range-0100/)

Domain, codomain, and range · qf-density15-typed-mapping-table-candidate-audit · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261011-advance-20261015-r03-domain-codomain-range-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261011-advance-20261015-r03-domain-codomain-range-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261011-advance-20261015-r03-domain-codomain-range-foundations/validation-report.json)