Complete Typed Function Data and Table Protocols
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Domain, codomain, and range · qf-density15-typed-mapping-table-candidate-audit · candidate
Canonical question JSONL · English JSONL · Validation report