Typed Relations, Projections, and Totality Guards
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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-20261012-advance-20261017-r03-relation-foundations · 2026-10-12 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Typed Relation Projections and Missing Sources
The explicitly typed relation R⊆Fin(0)×Fin(0) has ordered pairs=[]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).
View answer and explanation
([0, 0], [], [], [], 0, [])
Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0], [], [], [], 0, []).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-full-typed-inventory · candidate
2. Typed Relation Projections and Missing Sources
The explicitly typed relation R⊆Fin(2)×Fin(0) has ordered pairs=[]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).
View answer and explanation
([2, 0], [], [], [], 0, [0, 1])
Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0], [], [], [], 0, [0, 1]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-full-typed-inventory · candidate
3. Typed Relation Projections and Missing Sources
The explicitly typed relation R⊆Fin(0)×Fin(2) has ordered pairs=[]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).
View answer and explanation
([0, 2], [], [], [], 0, [])
Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2], [], [], [], 0, []).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-full-typed-inventory · candidate
4. Typed Relation Projections and Missing Sources
The explicitly typed relation R⊆Fin(2)×Fin(2) has ordered pairs=[[0, 0], [1, 1], [0, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).
View answer and explanation
([2, 2], [[0, 0], [1, 1]], [0, 1], [0, 1], 2, [])
Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2], [[0, 0], [1, 1]], [0, 1], [0, 1], 2, []).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-full-typed-inventory · candidate
5. Typed Relation Projections and Missing Sources
The explicitly typed relation R⊆Fin(2)×Fin(2) has ordered pairs=[[0, 0], [0, 1]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).
View answer and explanation
([2, 2], [[0, 0], [0, 1]], [0], [0, 1], 2, [1])
Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2], [[0, 0], [0, 1]], [0], [0, 1], 2, [1]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-full-typed-inventory · candidate
6. Typed Relation Projections and Missing Sources
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).
View answer and explanation
([3, 3], [[0, 0]], [0], [0], 1, [1, 2])
Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 0]], [0], [0], 1, [1, 2]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-full-typed-inventory · candidate
7. Typed Relation Projections and Missing Sources
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 1]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).
View answer and explanation
([3, 3], [[0, 1]], [0], [1], 1, [1, 2])
Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 1]], [0], [1], 1, [1, 2]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-full-typed-inventory · candidate
8. Typed Relation Projections and Missing Sources
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 1]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).
View answer and explanation
([3, 3], [[0, 0], [0, 1]], [0], [0, 1], 2, [1, 2])
Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [0, 1]], [0], [0, 1], 2, [1, 2]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-full-typed-inventory · candidate
9. Typed Relation Projections and Missing Sources
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 2]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).
View answer and explanation
([3, 3], [[0, 2]], [0], [2], 1, [1, 2])
Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 2]], [0], [2], 1, [1, 2]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-full-typed-inventory · candidate
10. Typed Relation Projections and Missing Sources
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 2]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).
View answer and explanation
([3, 3], [[0, 0], [0, 2]], [0], [0, 2], 2, [1, 2])
Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [0, 2]], [0], [0, 2], 2, [1, 2]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-full-typed-inventory · candidate
11. Typed Relation Projections and Missing Sources
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 1], [0, 2]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).
View answer and explanation
([3, 3], [[0, 1], [0, 2]], [0], [1, 2], 2, [1, 2])
Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 1], [0, 2]], [0], [1, 2], 2, [1, 2]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-full-typed-inventory · candidate
12. Typed Relation Projections and Missing Sources
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 1], [0, 2]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).
View answer and explanation
([3, 3], [[0, 0], [0, 1], [0, 2]], [0], [0, 1, 2], 3, [1, 2])
Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [0, 1], [0, 2]], [0], [0, 1, 2], 3, [1, 2]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-full-typed-inventory · candidate
13. Typed Relation Projections and Missing Sources
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[1, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).
View answer and explanation
([3, 3], [[1, 0]], [1], [0], 1, [0, 2])
Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[1, 0]], [1], [0], 1, [0, 2]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-full-typed-inventory · candidate
14. Typed Relation Projections and Missing Sources
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [1, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).
View answer and explanation
([3, 3], [[0, 0], [1, 0]], [0, 1], [0], 2, [2])
Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [1, 0]], [0, 1], [0], 2, [2]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-full-typed-inventory · candidate
15. Typed Relation Projections and Missing Sources
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 1], [1, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).
View answer and explanation
([3, 3], [[0, 1], [1, 0]], [0, 1], [0, 1], 2, [2])
Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 1], [1, 0]], [0, 1], [0, 1], 2, [2]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-full-typed-inventory · candidate
16. Typed Relation Projections and Missing Sources
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 1], [1, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).
View answer and explanation
([3, 3], [[0, 0], [0, 1], [1, 0]], [0, 1], [0, 1], 3, [2])
Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [0, 1], [1, 0]], [0, 1], [0, 1], 3, [2]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-full-typed-inventory · candidate
17. Typed Relation Projections and Missing Sources
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 2], [1, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).
View answer and explanation
([3, 3], [[0, 2], [1, 0]], [0, 1], [0, 2], 2, [2])
Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 2], [1, 0]], [0, 1], [0, 2], 2, [2]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-full-typed-inventory · candidate
18. Typed Relation Projections and Missing Sources
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 2], [1, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).
View answer and explanation
([3, 3], [[0, 0], [0, 2], [1, 0]], [0, 1], [0, 2], 3, [2])
Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [0, 2], [1, 0]], [0, 1], [0, 2], 3, [2]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-full-typed-inventory · candidate
19. Typed Relation Projections and Missing Sources
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 1], [0, 2], [1, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).
View answer and explanation
([3, 3], [[0, 1], [0, 2], [1, 0]], [0, 1], [0, 1, 2], 3, [2])
Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 1], [0, 2], [1, 0]], [0, 1], [0, 1, 2], 3, [2]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-full-typed-inventory · candidate
20. Typed Relation Projections and Missing Sources
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 1], [0, 2], [1, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).
View answer and explanation
([3, 3], [[0, 0], [0, 1], [0, 2], [1, 0]], [0, 1], [0, 1, 2], 4, [2])
Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [0, 1], [0, 2], [1, 0]], [0, 1], [0, 1, 2], 4, [2]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-full-typed-inventory · candidate
21. Typed Relation Projections and Missing Sources
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[1, 1]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).
View answer and explanation
([3, 3], [[1, 1]], [1], [1], 1, [0, 2])
Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[1, 1]], [1], [1], 1, [0, 2]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-full-typed-inventory · candidate
22. Typed Relation Projections and Missing Sources
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [1, 1]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).
View answer and explanation
([3, 3], [[0, 0], [1, 1]], [0, 1], [0, 1], 2, [2])
Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [1, 1]], [0, 1], [0, 1], 2, [2]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-full-typed-inventory · candidate
23. Typed Relation Projections and Missing Sources
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 1], [1, 1]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).
View answer and explanation
([3, 3], [[0, 1], [1, 1]], [0, 1], [1], 2, [2])
Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 1], [1, 1]], [0, 1], [1], 2, [2]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-full-typed-inventory · candidate
24. Typed Relation Projections and Missing Sources
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 1], [1, 1]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).
View answer and explanation
([3, 3], [[0, 0], [0, 1], [1, 1]], [0, 1], [0, 1], 3, [2])
Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [0, 1], [1, 1]], [0, 1], [0, 1], 3, [2]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-full-typed-inventory · candidate
25. Typed Relation Projections and Missing Sources
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 2], [1, 1]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).
View answer and explanation
([3, 3], [[0, 2], [1, 1]], [0, 1], [1, 2], 2, [2])
Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 2], [1, 1]], [0, 1], [1, 2], 2, [2]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-full-typed-inventory · candidate
26. Restricted Relation Image and All Target Fibers
The explicitly typed relation R⊆Fin(0)×Fin(0) has ordered pairs=[]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. The source subset is []. Give (normalized pairs, sorted source subset, its image, all target fibers restricted to that subset). Normalize repeated subset labels as a set; do not renumber them.
View answer and explanation
([], [], [], [])
Retain only pairs whose original source lies in the declared subset and enumerate every declared target fiber. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [], [], []).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-subset-image-fibers · candidate
27. Restricted Relation Image and All Target Fibers
The explicitly typed relation R⊆Fin(2)×Fin(2) has ordered pairs=[]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. The source subset is [1]. Give (normalized pairs, sorted source subset, its image, all target fibers restricted to that subset). Normalize repeated subset labels as a set; do not renumber them.
View answer and explanation
([], [1], [], [[], []])
Retain only pairs whose original source lies in the declared subset and enumerate every declared target fiber. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [1], [], [[], []]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-subset-image-fibers · candidate
28. Restricted Relation Image and All Target Fibers
The explicitly typed relation R⊆Fin(2)×Fin(2) has ordered pairs=[[0, 0], [0, 1], [1, 1]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. The source subset is []. Give (normalized pairs, sorted source subset, its image, all target fibers restricted to that subset). Normalize repeated subset labels as a set; do not renumber them.
View answer and explanation
([[0, 0], [0, 1], [1, 1]], [], [], [[], []])
Retain only pairs whose original source lies in the declared subset and enumerate every declared target fiber. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 0], [0, 1], [1, 1]], [], [], [[], []]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-subset-image-fibers · candidate
29. Restricted Relation Image and All Target Fibers
The explicitly typed relation R⊆Fin(2)×Fin(2) has ordered pairs=[[0, 0], [0, 1], [1, 1]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. The source subset is [0]. Give (normalized pairs, sorted source subset, its image, all target fibers restricted to that subset). Normalize repeated subset labels as a set; do not renumber them.
View answer and explanation
([[0, 0], [0, 1], [1, 1]], [0], [0, 1], [[0], [0]])
Retain only pairs whose original source lies in the declared subset and enumerate every declared target fiber. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 0], [0, 1], [1, 1]], [0], [0, 1], [[0], [0]]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-subset-image-fibers · candidate
30. Restricted Relation Image and All Target Fibers
The explicitly typed relation R⊆Fin(2)×Fin(2) has ordered pairs=[[0, 0], [1, 1]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. The source subset is [1]. Give (normalized pairs, sorted source subset, its image, all target fibers restricted to that subset). Normalize repeated subset labels as a set; do not renumber them.
View answer and explanation
([[0, 0], [1, 1]], [1], [1], [[], [1]])
Retain only pairs whose original source lies in the declared subset and enumerate every declared target fiber. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 0], [1, 1]], [1], [1], [[], [1]]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-subset-image-fibers · candidate
31. Restricted Relation Image and All Target Fibers
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. The source subset is [2, 0]. Give (normalized pairs, sorted source subset, its image, all target fibers restricted to that subset). Normalize repeated subset labels as a set; do not renumber them.
View answer and explanation
([[0, 0]], [0, 2], [0], [[0], [], []])
Retain only pairs whose original source lies in the declared subset and enumerate every declared target fiber. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 0]], [0, 2], [0], [[0], [], []]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-subset-image-fibers · candidate
32. Restricted Relation Image and All Target Fibers
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 1]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. The source subset is [2, 0]. Give (normalized pairs, sorted source subset, its image, all target fibers restricted to that subset). Normalize repeated subset labels as a set; do not renumber them.
View answer and explanation
([[0, 1]], [0, 2], [1], [[], [0], []])
Retain only pairs whose original source lies in the declared subset and enumerate every declared target fiber. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1]], [0, 2], [1], [[], [0], []]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-subset-image-fibers · candidate
33. Restricted Relation Image and All Target Fibers
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 1]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. The source subset is [2, 0]. Give (normalized pairs, sorted source subset, its image, all target fibers restricted to that subset). Normalize repeated subset labels as a set; do not renumber them.
View answer and explanation
([[0, 0], [0, 1]], [0, 2], [0, 1], [[0], [0], []])
Retain only pairs whose original source lies in the declared subset and enumerate every declared target fiber. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 0], [0, 1]], [0, 2], [0, 1], [[0], [0], []]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-subset-image-fibers · candidate
34. Restricted Relation Image and All Target Fibers
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 2]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. The source subset is [2, 0]. Give (normalized pairs, sorted source subset, its image, all target fibers restricted to that subset). Normalize repeated subset labels as a set; do not renumber them.
View answer and explanation
([[0, 2]], [0, 2], [2], [[], [], [0]])
Retain only pairs whose original source lies in the declared subset and enumerate every declared target fiber. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 2]], [0, 2], [2], [[], [], [0]]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-subset-image-fibers · candidate
35. Restricted Relation Image and All Target Fibers
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 2]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. The source subset is [2, 0]. Give (normalized pairs, sorted source subset, its image, all target fibers restricted to that subset). Normalize repeated subset labels as a set; do not renumber them.
View answer and explanation
([[0, 0], [0, 2]], [0, 2], [0, 2], [[0], [], [0]])
Retain only pairs whose original source lies in the declared subset and enumerate every declared target fiber. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 0], [0, 2]], [0, 2], [0, 2], [[0], [], [0]]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-subset-image-fibers · candidate
36. Restricted Relation Image and All Target Fibers
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 1], [0, 2]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. The source subset is [2, 0]. Give (normalized pairs, sorted source subset, its image, all target fibers restricted to that subset). Normalize repeated subset labels as a set; do not renumber them.
View answer and explanation
([[0, 1], [0, 2]], [0, 2], [1, 2], [[], [0], [0]])
Retain only pairs whose original source lies in the declared subset and enumerate every declared target fiber. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [0, 2]], [0, 2], [1, 2], [[], [0], [0]]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-subset-image-fibers · candidate
37. Restricted Relation Image and All Target Fibers
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 1], [0, 2]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. The source subset is [2, 0]. Give (normalized pairs, sorted source subset, its image, all target fibers restricted to that subset). Normalize repeated subset labels as a set; do not renumber them.
View answer and explanation
([[0, 0], [0, 1], [0, 2]], [0, 2], [0, 1, 2], [[0], [0], [0]])
Retain only pairs whose original source lies in the declared subset and enumerate every declared target fiber. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 0], [0, 1], [0, 2]], [0, 2], [0, 1, 2], [[0], [0], [0]]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-subset-image-fibers · candidate
38. Restricted Relation Image and All Target Fibers
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[1, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. The source subset is [2, 0]. Give (normalized pairs, sorted source subset, its image, all target fibers restricted to that subset). Normalize repeated subset labels as a set; do not renumber them.
View answer and explanation
([[1, 0]], [0, 2], [], [[], [], []])
Retain only pairs whose original source lies in the declared subset and enumerate every declared target fiber. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 0]], [0, 2], [], [[], [], []]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-subset-image-fibers · candidate
39. Restricted Relation Image and All Target Fibers
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [1, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. The source subset is [2, 0]. Give (normalized pairs, sorted source subset, its image, all target fibers restricted to that subset). Normalize repeated subset labels as a set; do not renumber them.
View answer and explanation
([[0, 0], [1, 0]], [0, 2], [0], [[0], [], []])
Retain only pairs whose original source lies in the declared subset and enumerate every declared target fiber. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 0], [1, 0]], [0, 2], [0], [[0], [], []]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-subset-image-fibers · candidate
40. Restricted Relation Image and All Target Fibers
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 1], [1, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. The source subset is [2, 0]. Give (normalized pairs, sorted source subset, its image, all target fibers restricted to that subset). Normalize repeated subset labels as a set; do not renumber them.
View answer and explanation
([[0, 1], [1, 0]], [0, 2], [1], [[], [0], []])
Retain only pairs whose original source lies in the declared subset and enumerate every declared target fiber. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [1, 0]], [0, 2], [1], [[], [0], []]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-subset-image-fibers · candidate
41. Restricted Relation Image and All Target Fibers
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 1], [1, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. The source subset is [2, 0]. Give (normalized pairs, sorted source subset, its image, all target fibers restricted to that subset). Normalize repeated subset labels as a set; do not renumber them.
View answer and explanation
([[0, 0], [0, 1], [1, 0]], [0, 2], [0, 1], [[0], [0], []])
Retain only pairs whose original source lies in the declared subset and enumerate every declared target fiber. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 0], [0, 1], [1, 0]], [0, 2], [0, 1], [[0], [0], []]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-subset-image-fibers · candidate
42. Restricted Relation Image and All Target Fibers
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 2], [1, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. The source subset is [2, 0]. Give (normalized pairs, sorted source subset, its image, all target fibers restricted to that subset). Normalize repeated subset labels as a set; do not renumber them.
View answer and explanation
([[0, 2], [1, 0]], [0, 2], [2], [[], [], [0]])
Retain only pairs whose original source lies in the declared subset and enumerate every declared target fiber. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 2], [1, 0]], [0, 2], [2], [[], [], [0]]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-subset-image-fibers · candidate
43. Restricted Relation Image and All Target Fibers
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 2], [1, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. The source subset is [2, 0]. Give (normalized pairs, sorted source subset, its image, all target fibers restricted to that subset). Normalize repeated subset labels as a set; do not renumber them.
View answer and explanation
([[0, 0], [0, 2], [1, 0]], [0, 2], [0, 2], [[0], [], [0]])
Retain only pairs whose original source lies in the declared subset and enumerate every declared target fiber. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 0], [0, 2], [1, 0]], [0, 2], [0, 2], [[0], [], [0]]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-subset-image-fibers · candidate
44. Restricted Relation Image and All Target Fibers
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 1], [0, 2], [1, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. The source subset is [2, 0]. Give (normalized pairs, sorted source subset, its image, all target fibers restricted to that subset). Normalize repeated subset labels as a set; do not renumber them.
View answer and explanation
([[0, 1], [0, 2], [1, 0]], [0, 2], [1, 2], [[], [0], [0]])
Retain only pairs whose original source lies in the declared subset and enumerate every declared target fiber. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [0, 2], [1, 0]], [0, 2], [1, 2], [[], [0], [0]]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-subset-image-fibers · candidate
45. Restricted Relation Image and All Target Fibers
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 1], [0, 2], [1, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. The source subset is [2, 0]. Give (normalized pairs, sorted source subset, its image, all target fibers restricted to that subset). Normalize repeated subset labels as a set; do not renumber them.
View answer and explanation
([[0, 0], [0, 1], [0, 2], [1, 0]], [0, 2], [0, 1, 2], [[0], [0], [0]])
Retain only pairs whose original source lies in the declared subset and enumerate every declared target fiber. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 0], [0, 1], [0, 2], [1, 0]], [0, 2], [0, 1, 2], [[0], [0], [0]]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-subset-image-fibers · candidate
46. Restricted Relation Image and All Target Fibers
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[1, 1]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. The source subset is [2, 0]. Give (normalized pairs, sorted source subset, its image, all target fibers restricted to that subset). Normalize repeated subset labels as a set; do not renumber them.
View answer and explanation
([[1, 1]], [0, 2], [], [[], [], []])
Retain only pairs whose original source lies in the declared subset and enumerate every declared target fiber. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 1]], [0, 2], [], [[], [], []]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-subset-image-fibers · candidate
47. Restricted Relation Image and All Target Fibers
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [1, 1]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. The source subset is [2, 0]. Give (normalized pairs, sorted source subset, its image, all target fibers restricted to that subset). Normalize repeated subset labels as a set; do not renumber them.
View answer and explanation
([[0, 0], [1, 1]], [0, 2], [0], [[0], [], []])
Retain only pairs whose original source lies in the declared subset and enumerate every declared target fiber. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 0], [1, 1]], [0, 2], [0], [[0], [], []]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-subset-image-fibers · candidate
48. Restricted Relation Image and All Target Fibers
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 1], [1, 1]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. The source subset is [2, 0]. Give (normalized pairs, sorted source subset, its image, all target fibers restricted to that subset). Normalize repeated subset labels as a set; do not renumber them.
View answer and explanation
([[0, 1], [1, 1]], [0, 2], [1], [[], [0], []])
Retain only pairs whose original source lies in the declared subset and enumerate every declared target fiber. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [1, 1]], [0, 2], [1], [[], [0], []]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-subset-image-fibers · candidate
49. Restricted Relation Image and All Target Fibers
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 1], [1, 1]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. The source subset is [2, 0]. Give (normalized pairs, sorted source subset, its image, all target fibers restricted to that subset). Normalize repeated subset labels as a set; do not renumber them.
View answer and explanation
([[0, 0], [0, 1], [1, 1]], [0, 2], [0, 1], [[0], [0], []])
Retain only pairs whose original source lies in the declared subset and enumerate every declared target fiber. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 0], [0, 1], [1, 1]], [0, 2], [0, 1], [[0], [0], []]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-subset-image-fibers · candidate
50. Restricted Relation Image and All Target Fibers
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 2], [1, 1]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. The source subset is [2, 0]. Give (normalized pairs, sorted source subset, its image, all target fibers restricted to that subset). Normalize repeated subset labels as a set; do not renumber them.
View answer and explanation
([[0, 2], [1, 1]], [0, 2], [2], [[], [], [0]])
Retain only pairs whose original source lies in the declared subset and enumerate every declared target fiber. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 2], [1, 1]], [0, 2], [2], [[], [], [0]]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-subset-image-fibers · candidate
51. Typed Converse Relation and Double Converse
The explicitly typed relation R⊆Fin(0)×Fin(0) has ordered pairs=[]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).
View answer and explanation
([0, 0], [0, 0], [], [], [], Yes)
Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0], [0, 0], [], [], [], Yes).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-inverse-double · candidate
52. Typed Converse Relation and Double Converse
The explicitly typed relation R⊆Fin(2)×Fin(0) has ordered pairs=[]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).
View answer and explanation
([2, 0], [0, 2], [], [], [], Yes)
Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0], [0, 2], [], [], [], Yes).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-inverse-double · candidate
53. Typed Converse Relation and Double Converse
The explicitly typed relation R⊆Fin(0)×Fin(2) has ordered pairs=[]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).
View answer and explanation
([0, 2], [2, 0], [], [], [], Yes)
Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2], [2, 0], [], [], [], Yes).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-inverse-double · candidate
54. Typed Converse Relation and Double Converse
The explicitly typed relation R⊆Fin(2)×Fin(2) has ordered pairs=[[0, 0], [1, 1], [0, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).
View answer and explanation
([2, 2], [2, 2], [[0, 0], [1, 1]], [[0, 0], [1, 1]], [[0, 0], [1, 1]], Yes)
Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2], [2, 2], [[0, 0], [1, 1]], [[0, 0], [1, 1]], [[0, 0], [1, 1]], Yes).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-inverse-double · candidate
55. Typed Converse Relation and Double Converse
The explicitly typed relation R⊆Fin(2)×Fin(2) has ordered pairs=[[0, 0], [0, 1]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).
View answer and explanation
([2, 2], [2, 2], [[0, 0], [0, 1]], [[0, 0], [1, 0]], [[0, 0], [0, 1]], Yes)
Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2], [2, 2], [[0, 0], [0, 1]], [[0, 0], [1, 0]], [[0, 0], [0, 1]], Yes).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-inverse-double · candidate
56. Typed Converse Relation and Double Converse
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).
View answer and explanation
([3, 3], [3, 3], [[0, 0]], [[0, 0]], [[0, 0]], Yes)
Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [3, 3], [[0, 0]], [[0, 0]], [[0, 0]], Yes).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-inverse-double · candidate
57. Typed Converse Relation and Double Converse
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 1]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).
View answer and explanation
([3, 3], [3, 3], [[0, 1]], [[1, 0]], [[0, 1]], Yes)
Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [3, 3], [[0, 1]], [[1, 0]], [[0, 1]], Yes).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-inverse-double · candidate
58. Typed Converse Relation and Double Converse
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 1]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).
View answer and explanation
([3, 3], [3, 3], [[0, 0], [0, 1]], [[0, 0], [1, 0]], [[0, 0], [0, 1]], Yes)
Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [3, 3], [[0, 0], [0, 1]], [[0, 0], [1, 0]], [[0, 0], [0, 1]], Yes).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-inverse-double · candidate
59. Typed Converse Relation and Double Converse
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 2]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).
View answer and explanation
([3, 3], [3, 3], [[0, 2]], [[2, 0]], [[0, 2]], Yes)
Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [3, 3], [[0, 2]], [[2, 0]], [[0, 2]], Yes).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-inverse-double · candidate
60. Typed Converse Relation and Double Converse
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 2]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).
View answer and explanation
([3, 3], [3, 3], [[0, 0], [0, 2]], [[0, 0], [2, 0]], [[0, 0], [0, 2]], Yes)
Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [3, 3], [[0, 0], [0, 2]], [[0, 0], [2, 0]], [[0, 0], [0, 2]], Yes).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-inverse-double · candidate
61. Typed Converse Relation and Double Converse
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 1], [0, 2]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).
View answer and explanation
([3, 3], [3, 3], [[0, 1], [0, 2]], [[1, 0], [2, 0]], [[0, 1], [0, 2]], Yes)
Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [3, 3], [[0, 1], [0, 2]], [[1, 0], [2, 0]], [[0, 1], [0, 2]], Yes).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-inverse-double · candidate
62. Typed Converse Relation and Double Converse
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 1], [0, 2]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).
View answer and explanation
([3, 3], [3, 3], [[0, 0], [0, 1], [0, 2]], [[0, 0], [1, 0], [2, 0]], [[0, 0], [0, 1], [0, 2]], Yes)
Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [3, 3], [[0, 0], [0, 1], [0, 2]], [[0, 0], [1, 0], [2, 0]], [[0, 0], [0, 1], [0, 2]], Yes).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-inverse-double · candidate
63. Typed Converse Relation and Double Converse
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[1, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).
View answer and explanation
([3, 3], [3, 3], [[1, 0]], [[0, 1]], [[1, 0]], Yes)
Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [3, 3], [[1, 0]], [[0, 1]], [[1, 0]], Yes).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-inverse-double · candidate
64. Typed Converse Relation and Double Converse
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [1, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).
View answer and explanation
([3, 3], [3, 3], [[0, 0], [1, 0]], [[0, 0], [0, 1]], [[0, 0], [1, 0]], Yes)
Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [3, 3], [[0, 0], [1, 0]], [[0, 0], [0, 1]], [[0, 0], [1, 0]], Yes).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-inverse-double · candidate
65. Typed Converse Relation and Double Converse
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 1], [1, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).
View answer and explanation
([3, 3], [3, 3], [[0, 1], [1, 0]], [[0, 1], [1, 0]], [[0, 1], [1, 0]], Yes)
Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [3, 3], [[0, 1], [1, 0]], [[0, 1], [1, 0]], [[0, 1], [1, 0]], Yes).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-inverse-double · candidate
66. Typed Converse Relation and Double Converse
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 1], [1, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).
View answer and explanation
([3, 3], [3, 3], [[0, 0], [0, 1], [1, 0]], [[0, 0], [0, 1], [1, 0]], [[0, 0], [0, 1], [1, 0]], Yes)
Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [3, 3], [[0, 0], [0, 1], [1, 0]], [[0, 0], [0, 1], [1, 0]], [[0, 0], [0, 1], [1, 0]], Yes).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-inverse-double · candidate
67. Typed Converse Relation and Double Converse
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 2], [1, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).
View answer and explanation
([3, 3], [3, 3], [[0, 2], [1, 0]], [[0, 1], [2, 0]], [[0, 2], [1, 0]], Yes)
Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [3, 3], [[0, 2], [1, 0]], [[0, 1], [2, 0]], [[0, 2], [1, 0]], Yes).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-inverse-double · candidate
68. Typed Converse Relation and Double Converse
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 2], [1, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).
View answer and explanation
([3, 3], [3, 3], [[0, 0], [0, 2], [1, 0]], [[0, 0], [0, 1], [2, 0]], [[0, 0], [0, 2], [1, 0]], Yes)
Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [3, 3], [[0, 0], [0, 2], [1, 0]], [[0, 0], [0, 1], [2, 0]], [[0, 0], [0, 2], [1, 0]], Yes).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-inverse-double · candidate
69. Typed Converse Relation and Double Converse
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 1], [0, 2], [1, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).
View answer and explanation
([3, 3], [3, 3], [[0, 1], [0, 2], [1, 0]], [[0, 1], [1, 0], [2, 0]], [[0, 1], [0, 2], [1, 0]], Yes)
Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [3, 3], [[0, 1], [0, 2], [1, 0]], [[0, 1], [1, 0], [2, 0]], [[0, 1], [0, 2], [1, 0]], Yes).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-inverse-double · candidate
70. Typed Converse Relation and Double Converse
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 1], [0, 2], [1, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).
View answer and explanation
([3, 3], [3, 3], [[0, 0], [0, 1], [0, 2], [1, 0]], [[0, 0], [0, 1], [1, 0], [2, 0]], [[0, 0], [0, 1], [0, 2], [1, 0]], Yes)
Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [3, 3], [[0, 0], [0, 1], [0, 2], [1, 0]], [[0, 0], [0, 1], [1, 0], [2, 0]], [[0, 0], [0, 1], [0, 2], [1, 0]], Yes).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-inverse-double · candidate
71. Typed Converse Relation and Double Converse
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[1, 1]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).
View answer and explanation
([3, 3], [3, 3], [[1, 1]], [[1, 1]], [[1, 1]], Yes)
Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [3, 3], [[1, 1]], [[1, 1]], [[1, 1]], Yes).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-inverse-double · candidate
72. Typed Converse Relation and Double Converse
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [1, 1]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).
View answer and explanation
([3, 3], [3, 3], [[0, 0], [1, 1]], [[0, 0], [1, 1]], [[0, 0], [1, 1]], Yes)
Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [3, 3], [[0, 0], [1, 1]], [[0, 0], [1, 1]], [[0, 0], [1, 1]], Yes).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-inverse-double · candidate
73. Typed Converse Relation and Double Converse
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 1], [1, 1]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).
View answer and explanation
([3, 3], [3, 3], [[0, 1], [1, 1]], [[1, 0], [1, 1]], [[0, 1], [1, 1]], Yes)
Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [3, 3], [[0, 1], [1, 1]], [[1, 0], [1, 1]], [[0, 1], [1, 1]], Yes).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-inverse-double · candidate
74. Typed Converse Relation and Double Converse
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 1], [1, 1]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).
View answer and explanation
([3, 3], [3, 3], [[0, 0], [0, 1], [1, 1]], [[0, 0], [1, 0], [1, 1]], [[0, 0], [0, 1], [1, 1]], Yes)
Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [3, 3], [[0, 0], [0, 1], [1, 1]], [[0, 0], [1, 0], [1, 1]], [[0, 0], [0, 1], [1, 1]], Yes).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-inverse-double · candidate
75. Typed Converse Relation and Double Converse
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 2], [1, 1]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).
View answer and explanation
([3, 3], [3, 3], [[0, 2], [1, 1]], [[1, 1], [2, 0]], [[0, 2], [1, 1]], Yes)
Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [3, 3], [[0, 2], [1, 1]], [[1, 1], [2, 0]], [[0, 2], [1, 1]], Yes).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-inverse-double · candidate
76. Relation Single-Valuedness versus Total-Function Qualification
The explicitly typed relation R⊆Fin(0)×Fin(0) has ordered pairs=[]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).
View answer and explanation
([0, 0], [], [], Yes, Yes, [])
At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0], [], [], Yes, Yes, []).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-single-valued-total · candidate
77. Relation Single-Valuedness versus Total-Function Qualification
The explicitly typed relation R⊆Fin(2)×Fin(0) has ordered pairs=[]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).
View answer and explanation
([2, 0], [], [0, 0], Yes, No, not-a-total-function)
At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0], [], [0, 0], Yes, No, not-a-total-function).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-single-valued-total · candidate
78. Relation Single-Valuedness versus Total-Function Qualification
The explicitly typed relation R⊆Fin(0)×Fin(2) has ordered pairs=[]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).
View answer and explanation
([0, 2], [], [], Yes, Yes, [])
At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2], [], [], Yes, Yes, []).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-single-valued-total · candidate
79. Relation Single-Valuedness versus Total-Function Qualification
The explicitly typed relation R⊆Fin(2)×Fin(2) has ordered pairs=[[0, 0], [1, 1], [0, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).
View answer and explanation
([2, 2], [[0, 0], [1, 1]], [1, 1], Yes, Yes, [0, 1])
At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2], [[0, 0], [1, 1]], [1, 1], Yes, Yes, [0, 1]).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-single-valued-total · candidate
80. Relation Single-Valuedness versus Total-Function Qualification
The explicitly typed relation R⊆Fin(2)×Fin(2) has ordered pairs=[[0, 0], [0, 1]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).
View answer and explanation
([2, 2], [[0, 0], [0, 1]], [2, 0], No, No, not-a-total-function)
At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2], [[0, 0], [0, 1]], [2, 0], No, No, not-a-total-function).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-single-valued-total · candidate
81. Relation Single-Valuedness versus Total-Function Qualification
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).
View answer and explanation
([3, 3], [[0, 0]], [1, 0, 0], Yes, No, not-a-total-function)
At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 0]], [1, 0, 0], Yes, No, not-a-total-function).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-single-valued-total · candidate
82. Relation Single-Valuedness versus Total-Function Qualification
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 1]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).
View answer and explanation
([3, 3], [[0, 1]], [1, 0, 0], Yes, No, not-a-total-function)
At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 1]], [1, 0, 0], Yes, No, not-a-total-function).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-single-valued-total · candidate
83. Relation Single-Valuedness versus Total-Function Qualification
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 1]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).
View answer and explanation
([3, 3], [[0, 0], [0, 1]], [2, 0, 0], No, No, not-a-total-function)
At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [0, 1]], [2, 0, 0], No, No, not-a-total-function).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-single-valued-total · candidate
84. Relation Single-Valuedness versus Total-Function Qualification
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 2]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).
View answer and explanation
([3, 3], [[0, 2]], [1, 0, 0], Yes, No, not-a-total-function)
At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 2]], [1, 0, 0], Yes, No, not-a-total-function).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-single-valued-total · candidate
85. Relation Single-Valuedness versus Total-Function Qualification
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 2]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).
View answer and explanation
([3, 3], [[0, 0], [0, 2]], [2, 0, 0], No, No, not-a-total-function)
At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [0, 2]], [2, 0, 0], No, No, not-a-total-function).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-single-valued-total · candidate
86. Relation Single-Valuedness versus Total-Function Qualification
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 1], [0, 2]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).
View answer and explanation
([3, 3], [[0, 1], [0, 2]], [2, 0, 0], No, No, not-a-total-function)
At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 1], [0, 2]], [2, 0, 0], No, No, not-a-total-function).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-single-valued-total · candidate
87. Relation Single-Valuedness versus Total-Function Qualification
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 1], [0, 2]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).
View answer and explanation
([3, 3], [[0, 0], [0, 1], [0, 2]], [3, 0, 0], No, No, not-a-total-function)
At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [0, 1], [0, 2]], [3, 0, 0], No, No, not-a-total-function).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-single-valued-total · candidate
88. Relation Single-Valuedness versus Total-Function Qualification
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[1, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).
View answer and explanation
([3, 3], [[1, 0]], [0, 1, 0], Yes, No, not-a-total-function)
At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[1, 0]], [0, 1, 0], Yes, No, not-a-total-function).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-single-valued-total · candidate
89. Relation Single-Valuedness versus Total-Function Qualification
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [1, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).
View answer and explanation
([3, 3], [[0, 0], [1, 0]], [1, 1, 0], Yes, No, not-a-total-function)
At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [1, 0]], [1, 1, 0], Yes, No, not-a-total-function).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-single-valued-total · candidate
90. Relation Single-Valuedness versus Total-Function Qualification
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 1], [1, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).
View answer and explanation
([3, 3], [[0, 1], [1, 0]], [1, 1, 0], Yes, No, not-a-total-function)
At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 1], [1, 0]], [1, 1, 0], Yes, No, not-a-total-function).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-single-valued-total · candidate
91. Relation Single-Valuedness versus Total-Function Qualification
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 1], [1, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).
View answer and explanation
([3, 3], [[0, 0], [0, 1], [1, 0]], [2, 1, 0], No, No, not-a-total-function)
At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [0, 1], [1, 0]], [2, 1, 0], No, No, not-a-total-function).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-single-valued-total · candidate
92. Relation Single-Valuedness versus Total-Function Qualification
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 2], [1, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).
View answer and explanation
([3, 3], [[0, 2], [1, 0]], [1, 1, 0], Yes, No, not-a-total-function)
At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 2], [1, 0]], [1, 1, 0], Yes, No, not-a-total-function).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-single-valued-total · candidate
93. Relation Single-Valuedness versus Total-Function Qualification
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 2], [1, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).
View answer and explanation
([3, 3], [[0, 0], [0, 2], [1, 0]], [2, 1, 0], No, No, not-a-total-function)
At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [0, 2], [1, 0]], [2, 1, 0], No, No, not-a-total-function).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-single-valued-total · candidate
94. Relation Single-Valuedness versus Total-Function Qualification
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 1], [0, 2], [1, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).
View answer and explanation
([3, 3], [[0, 1], [0, 2], [1, 0]], [2, 1, 0], No, No, not-a-total-function)
At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 1], [0, 2], [1, 0]], [2, 1, 0], No, No, not-a-total-function).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-single-valued-total · candidate
95. Relation Single-Valuedness versus Total-Function Qualification
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 1], [0, 2], [1, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).
View answer and explanation
([3, 3], [[0, 0], [0, 1], [0, 2], [1, 0]], [3, 1, 0], No, No, not-a-total-function)
At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [0, 1], [0, 2], [1, 0]], [3, 1, 0], No, No, not-a-total-function).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-single-valued-total · candidate
96. Relation Single-Valuedness versus Total-Function Qualification
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[1, 1]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).
View answer and explanation
([3, 3], [[1, 1]], [0, 1, 0], Yes, No, not-a-total-function)
At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[1, 1]], [0, 1, 0], Yes, No, not-a-total-function).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-single-valued-total · candidate
97. Relation Single-Valuedness versus Total-Function Qualification
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [1, 1]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).
View answer and explanation
([3, 3], [[0, 0], [1, 1]], [1, 1, 0], Yes, No, not-a-total-function)
At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [1, 1]], [1, 1, 0], Yes, No, not-a-total-function).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-single-valued-total · candidate
98. Relation Single-Valuedness versus Total-Function Qualification
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 1], [1, 1]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).
View answer and explanation
([3, 3], [[0, 1], [1, 1]], [1, 1, 0], Yes, No, not-a-total-function)
At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 1], [1, 1]], [1, 1, 0], Yes, No, not-a-total-function).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-single-valued-total · candidate
99. Relation Single-Valuedness versus Total-Function Qualification
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 1], [1, 1]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).
View answer and explanation
([3, 3], [[0, 0], [0, 1], [1, 1]], [2, 1, 0], No, No, not-a-total-function)
At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [0, 1], [1, 1]], [2, 1, 0], No, No, not-a-total-function).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-single-valued-total · candidate
100. Relation Single-Valuedness versus Total-Function Qualification
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 2], [1, 1]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).
View answer and explanation
([3, 3], [[0, 2], [1, 1]], [1, 1, 0], Yes, No, not-a-total-function)
At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 2], [1, 1]], [1, 1, 0], Yes, No, not-a-total-function).
Reviewed step-by-step derivation
Binary relations and set projections · qf-density17-relation-single-valued-total · candidate
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