# Typed Relations, Projections, and Totality Guards

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

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

batch-20261012-advance-20261017-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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0001/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0002/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0003/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0004/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0005/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0006/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0007/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0008/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0009/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0010/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0011/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0012/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0013/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0014/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0015/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0016/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0017/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0018/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0019/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0020/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0021/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0022/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0023/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0024/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0025/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0026/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0027/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0028/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0029/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0030/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0031/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0032/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0033/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0034/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0035/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0036/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0037/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0038/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0039/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0040/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0041/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0042/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0043/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0044/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0045/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0046/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0047/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0048/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0049/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0050/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0051/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0052/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0053/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0054/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0055/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0056/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0057/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0058/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0059/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0060/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0061/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0062/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0063/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0064/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0065/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0066/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0067/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0068/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0069/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0070/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0071/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0072/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0073/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0074/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0075/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0076/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0077/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0078/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0079/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0080/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0081/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0082/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0083/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0084/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0085/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0086/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0087/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0088/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0089/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0090/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0091/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0092/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0093/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0094/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0095/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0096/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0097/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0098/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0099/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r03-relation-0100/)

Binary relations and set projections · qf-density17-relation-single-valued-total · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261012-advance-20261017-r03-relation-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261012-advance-20261017-r03-relation-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261012-advance-20261017-r03-relation-foundations/validation-report.json)