OCMEBilingual mathematics & derivations繁體中文

Complete Finite-Map Bijections and Cardinality Guards

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

batch-20261012-advance-20261017-r01-bijective-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. Complete Fibers, Collisions, Missing Targets, and Map Class

Use the complete total map Fin(0)=∅→Fin(0) with images=[] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, every codomain fiber, all lexicographically ordered distinct-input collision pairs, missing targets, injectivity, surjectivity, bijectivity).

View answer and explanation

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

Enumerate all declared targets, including empty fibers, and keep the three function properties separate. The empty-to-empty map is bijective. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [], [], [], Yes, Yes, Yes).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-fibers-and-class · candidate

2. Complete Fibers, Collisions, Missing Targets, and Map Class

Use the complete total map Fin(0)=∅→Fin(2) with images=[] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, every codomain fiber, all lexicographically ordered distinct-input collision pairs, missing targets, injectivity, surjectivity, bijectivity).

View answer and explanation

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

Enumerate all declared targets, including empty fibers, and keep the three function properties separate. The empty-to-empty map is bijective. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [[], []], [], [0, 1], Yes, No, No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-fibers-and-class · candidate

3. Complete Fibers, Collisions, Missing Targets, and Map Class

Use the complete total map Fin(1)={0,…,0}→Fin(1) with images=[0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, every codomain fiber, all lexicographically ordered distinct-input collision pairs, missing targets, injectivity, surjectivity, bijectivity).

View answer and explanation

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

Enumerate all declared targets, including empty fibers, and keep the three function properties separate. The empty-to-empty map is bijective. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0], [[0]], [], [], Yes, Yes, Yes).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-fibers-and-class · candidate

4. Complete Fibers, Collisions, Missing Targets, and Map Class

Use the complete total map Fin(3)={0,…,2}→Fin(3) with images=[0, 1, 2] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, every codomain fiber, all lexicographically ordered distinct-input collision pairs, missing targets, injectivity, surjectivity, bijectivity).

View answer and explanation

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

Enumerate all declared targets, including empty fibers, and keep the three function properties separate. The empty-to-empty map is bijective. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2], [[0], [1], [2]], [], [], Yes, Yes, Yes).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-fibers-and-class · candidate

5. Complete Fibers, Collisions, Missing Targets, and Map Class

Use the complete total map Fin(3)={0,…,2}→Fin(3) with images=[0, 0, 1] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, every codomain fiber, all lexicographically ordered distinct-input collision pairs, missing targets, injectivity, surjectivity, bijectivity).

View answer and explanation

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

Enumerate all declared targets, including empty fibers, and keep the three function properties separate. The empty-to-empty map is bijective. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1], [[0, 1], [2], []], [[0, 1]], [2], No, No, No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-fibers-and-class · candidate

6. Complete Fibers, Collisions, Missing Targets, and Map Class

Use the complete total map Fin(3)={0,…,2}→Fin(2) with images=[0, 1, 1] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, every codomain fiber, all lexicographically ordered distinct-input collision pairs, missing targets, injectivity, surjectivity, bijectivity).

View answer and explanation

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

Enumerate all declared targets, including empty fibers, and keep the three function properties separate. The empty-to-empty map is bijective. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 1], [[0], [1, 2]], [[1, 2]], [], No, Yes, No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-fibers-and-class · candidate

7. Complete Fibers, Collisions, Missing Targets, and Map Class

Use the complete total map Fin(2)={0,…,1}→Fin(3) with images=[0, 1] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, every codomain fiber, all lexicographically ordered distinct-input collision pairs, missing targets, injectivity, surjectivity, bijectivity).

View answer and explanation

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

Enumerate all declared targets, including empty fibers, and keep the three function properties separate. The empty-to-empty map is bijective. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1], [[0], [1], []], [], [2], Yes, No, No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-fibers-and-class · candidate

8. Complete Fibers, Collisions, Missing Targets, and Map Class

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 0, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, every codomain fiber, all lexicographically ordered distinct-input collision pairs, missing targets, injectivity, surjectivity, bijectivity).

View answer and explanation

([0, 0, 0, 0, 0], [[0, 1, 2, 3, 4], [], []], [[0, 1], [0, 2], [0, 3], [0, 4], [1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]], [1, 2], No, No, No)

Enumerate all declared targets, including empty fibers, and keep the three function properties separate. The empty-to-empty map is bijective. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 0, 0, 0], [[0, 1, 2, 3, 4], [], []], [[0, 1], [0, 2], [0, 3], [0, 4], [1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]], [1, 2], No, No, No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-fibers-and-class · candidate

9. Complete Fibers, Collisions, Missing Targets, and Map Class

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 0, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, every codomain fiber, all lexicographically ordered distinct-input collision pairs, missing targets, injectivity, surjectivity, bijectivity).

View answer and explanation

([1, 0, 0, 0, 0], [[1, 2, 3, 4], [0], []], [[1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]], [2], No, No, No)

Enumerate all declared targets, including empty fibers, and keep the three function properties separate. The empty-to-empty map is bijective. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 0, 0, 0], [[1, 2, 3, 4], [0], []], [[1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]], [2], No, No, No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-fibers-and-class · candidate

10. Complete Fibers, Collisions, Missing Targets, and Map Class

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 0, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, every codomain fiber, all lexicographically ordered distinct-input collision pairs, missing targets, injectivity, surjectivity, bijectivity).

View answer and explanation

([2, 0, 0, 0, 0], [[1, 2, 3, 4], [], [0]], [[1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]], [1], No, No, No)

Enumerate all declared targets, including empty fibers, and keep the three function properties separate. The empty-to-empty map is bijective. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 0, 0, 0], [[1, 2, 3, 4], [], [0]], [[1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]], [1], No, No, No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-fibers-and-class · candidate

11. Complete Fibers, Collisions, Missing Targets, and Map Class

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 1, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, every codomain fiber, all lexicographically ordered distinct-input collision pairs, missing targets, injectivity, surjectivity, bijectivity).

View answer and explanation

([0, 1, 0, 0, 0], [[0, 2, 3, 4], [1], []], [[0, 2], [0, 3], [0, 4], [2, 3], [2, 4], [3, 4]], [2], No, No, No)

Enumerate all declared targets, including empty fibers, and keep the three function properties separate. The empty-to-empty map is bijective. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 0, 0, 0], [[0, 2, 3, 4], [1], []], [[0, 2], [0, 3], [0, 4], [2, 3], [2, 4], [3, 4]], [2], No, No, No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-fibers-and-class · candidate

12. Complete Fibers, Collisions, Missing Targets, and Map Class

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 1, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, every codomain fiber, all lexicographically ordered distinct-input collision pairs, missing targets, injectivity, surjectivity, bijectivity).

View answer and explanation

([1, 1, 0, 0, 0], [[2, 3, 4], [0, 1], []], [[0, 1], [2, 3], [2, 4], [3, 4]], [2], No, No, No)

Enumerate all declared targets, including empty fibers, and keep the three function properties separate. The empty-to-empty map is bijective. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 0, 0, 0], [[2, 3, 4], [0, 1], []], [[0, 1], [2, 3], [2, 4], [3, 4]], [2], No, No, No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-fibers-and-class · candidate

13. Complete Fibers, Collisions, Missing Targets, and Map Class

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 1, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, every codomain fiber, all lexicographically ordered distinct-input collision pairs, missing targets, injectivity, surjectivity, bijectivity).

View answer and explanation

([2, 1, 0, 0, 0], [[2, 3, 4], [1], [0]], [[2, 3], [2, 4], [3, 4]], [], No, Yes, No)

Enumerate all declared targets, including empty fibers, and keep the three function properties separate. The empty-to-empty map is bijective. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 0, 0, 0], [[2, 3, 4], [1], [0]], [[2, 3], [2, 4], [3, 4]], [], No, Yes, No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-fibers-and-class · candidate

14. Complete Fibers, Collisions, Missing Targets, and Map Class

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 2, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, every codomain fiber, all lexicographically ordered distinct-input collision pairs, missing targets, injectivity, surjectivity, bijectivity).

View answer and explanation

([0, 2, 0, 0, 0], [[0, 2, 3, 4], [], [1]], [[0, 2], [0, 3], [0, 4], [2, 3], [2, 4], [3, 4]], [1], No, No, No)

Enumerate all declared targets, including empty fibers, and keep the three function properties separate. The empty-to-empty map is bijective. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 0, 0, 0], [[0, 2, 3, 4], [], [1]], [[0, 2], [0, 3], [0, 4], [2, 3], [2, 4], [3, 4]], [1], No, No, No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-fibers-and-class · candidate

15. Complete Fibers, Collisions, Missing Targets, and Map Class

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 2, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, every codomain fiber, all lexicographically ordered distinct-input collision pairs, missing targets, injectivity, surjectivity, bijectivity).

View answer and explanation

([1, 2, 0, 0, 0], [[2, 3, 4], [0], [1]], [[2, 3], [2, 4], [3, 4]], [], No, Yes, No)

Enumerate all declared targets, including empty fibers, and keep the three function properties separate. The empty-to-empty map is bijective. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 0, 0, 0], [[2, 3, 4], [0], [1]], [[2, 3], [2, 4], [3, 4]], [], No, Yes, No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-fibers-and-class · candidate

16. Complete Fibers, Collisions, Missing Targets, and Map Class

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 2, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, every codomain fiber, all lexicographically ordered distinct-input collision pairs, missing targets, injectivity, surjectivity, bijectivity).

View answer and explanation

([2, 2, 0, 0, 0], [[2, 3, 4], [], [0, 1]], [[0, 1], [2, 3], [2, 4], [3, 4]], [1], No, No, No)

Enumerate all declared targets, including empty fibers, and keep the three function properties separate. The empty-to-empty map is bijective. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2, 0, 0, 0], [[2, 3, 4], [], [0, 1]], [[0, 1], [2, 3], [2, 4], [3, 4]], [1], No, No, No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-fibers-and-class · candidate

17. Complete Fibers, Collisions, Missing Targets, and Map Class

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 0, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, every codomain fiber, all lexicographically ordered distinct-input collision pairs, missing targets, injectivity, surjectivity, bijectivity).

View answer and explanation

([0, 0, 1, 0, 0], [[0, 1, 3, 4], [2], []], [[0, 1], [0, 3], [0, 4], [1, 3], [1, 4], [3, 4]], [2], No, No, No)

Enumerate all declared targets, including empty fibers, and keep the three function properties separate. The empty-to-empty map is bijective. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1, 0, 0], [[0, 1, 3, 4], [2], []], [[0, 1], [0, 3], [0, 4], [1, 3], [1, 4], [3, 4]], [2], No, No, No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-fibers-and-class · candidate

18. Complete Fibers, Collisions, Missing Targets, and Map Class

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 0, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, every codomain fiber, all lexicographically ordered distinct-input collision pairs, missing targets, injectivity, surjectivity, bijectivity).

View answer and explanation

([1, 0, 1, 0, 0], [[1, 3, 4], [0, 2], []], [[0, 2], [1, 3], [1, 4], [3, 4]], [2], No, No, No)

Enumerate all declared targets, including empty fibers, and keep the three function properties separate. The empty-to-empty map is bijective. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 1, 0, 0], [[1, 3, 4], [0, 2], []], [[0, 2], [1, 3], [1, 4], [3, 4]], [2], No, No, No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-fibers-and-class · candidate

19. Complete Fibers, Collisions, Missing Targets, and Map Class

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 0, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, every codomain fiber, all lexicographically ordered distinct-input collision pairs, missing targets, injectivity, surjectivity, bijectivity).

View answer and explanation

([2, 0, 1, 0, 0], [[1, 3, 4], [2], [0]], [[1, 3], [1, 4], [3, 4]], [], No, Yes, No)

Enumerate all declared targets, including empty fibers, and keep the three function properties separate. The empty-to-empty map is bijective. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 1, 0, 0], [[1, 3, 4], [2], [0]], [[1, 3], [1, 4], [3, 4]], [], No, Yes, No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-fibers-and-class · candidate

20. Complete Fibers, Collisions, Missing Targets, and Map Class

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 1, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, every codomain fiber, all lexicographically ordered distinct-input collision pairs, missing targets, injectivity, surjectivity, bijectivity).

View answer and explanation

([0, 1, 1, 0, 0], [[0, 3, 4], [1, 2], []], [[0, 3], [0, 4], [1, 2], [3, 4]], [2], No, No, No)

Enumerate all declared targets, including empty fibers, and keep the three function properties separate. The empty-to-empty map is bijective. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 1, 0, 0], [[0, 3, 4], [1, 2], []], [[0, 3], [0, 4], [1, 2], [3, 4]], [2], No, No, No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-fibers-and-class · candidate

21. Complete Fibers, Collisions, Missing Targets, and Map Class

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 1, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, every codomain fiber, all lexicographically ordered distinct-input collision pairs, missing targets, injectivity, surjectivity, bijectivity).

View answer and explanation

([1, 1, 1, 0, 0], [[3, 4], [0, 1, 2], []], [[0, 1], [0, 2], [1, 2], [3, 4]], [2], No, No, No)

Enumerate all declared targets, including empty fibers, and keep the three function properties separate. The empty-to-empty map is bijective. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 1, 0, 0], [[3, 4], [0, 1, 2], []], [[0, 1], [0, 2], [1, 2], [3, 4]], [2], No, No, No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-fibers-and-class · candidate

22. Complete Fibers, Collisions, Missing Targets, and Map Class

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 1, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, every codomain fiber, all lexicographically ordered distinct-input collision pairs, missing targets, injectivity, surjectivity, bijectivity).

View answer and explanation

([2, 1, 1, 0, 0], [[3, 4], [1, 2], [0]], [[1, 2], [3, 4]], [], No, Yes, No)

Enumerate all declared targets, including empty fibers, and keep the three function properties separate. The empty-to-empty map is bijective. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 1, 0, 0], [[3, 4], [1, 2], [0]], [[1, 2], [3, 4]], [], No, Yes, No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-fibers-and-class · candidate

23. Complete Fibers, Collisions, Missing Targets, and Map Class

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 2, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, every codomain fiber, all lexicographically ordered distinct-input collision pairs, missing targets, injectivity, surjectivity, bijectivity).

View answer and explanation

([0, 2, 1, 0, 0], [[0, 3, 4], [2], [1]], [[0, 3], [0, 4], [3, 4]], [], No, Yes, No)

Enumerate all declared targets, including empty fibers, and keep the three function properties separate. The empty-to-empty map is bijective. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 1, 0, 0], [[0, 3, 4], [2], [1]], [[0, 3], [0, 4], [3, 4]], [], No, Yes, No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-fibers-and-class · candidate

24. Complete Fibers, Collisions, Missing Targets, and Map Class

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 2, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, every codomain fiber, all lexicographically ordered distinct-input collision pairs, missing targets, injectivity, surjectivity, bijectivity).

View answer and explanation

([1, 2, 1, 0, 0], [[3, 4], [0, 2], [1]], [[0, 2], [3, 4]], [], No, Yes, No)

Enumerate all declared targets, including empty fibers, and keep the three function properties separate. The empty-to-empty map is bijective. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 1, 0, 0], [[3, 4], [0, 2], [1]], [[0, 2], [3, 4]], [], No, Yes, No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-fibers-and-class · candidate

25. Complete Fibers, Collisions, Missing Targets, and Map Class

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 2, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, every codomain fiber, all lexicographically ordered distinct-input collision pairs, missing targets, injectivity, surjectivity, bijectivity).

View answer and explanation

([2, 2, 1, 0, 0], [[3, 4], [2], [0, 1]], [[0, 1], [3, 4]], [], No, Yes, No)

Enumerate all declared targets, including empty fibers, and keep the three function properties separate. The empty-to-empty map is bijective. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2, 1, 0, 0], [[3, 4], [2], [0, 1]], [[0, 1], [3, 4]], [], No, Yes, No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-fibers-and-class · candidate

26. Converse Relation before Inverse-Function Qualification

Use the complete total map Fin(0)=∅→Fin(0) with images=[] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (sorted swapped relation pairs [f(x),x], each codomain fiber size, inverse-function eligibility, inverse table or not-an-inverse-function).

View answer and explanation

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

Swapping pairs creates a relation; it creates a total inverse function exactly when every target has one preimage. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [], Yes, []).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-swapped-relation-audit · candidate

27. Converse Relation before Inverse-Function Qualification

Use the complete total map Fin(0)=∅→Fin(2) with images=[] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (sorted swapped relation pairs [f(x),x], each codomain fiber size, inverse-function eligibility, inverse table or not-an-inverse-function).

View answer and explanation

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

Swapping pairs creates a relation; it creates a total inverse function exactly when every target has one preimage. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [0, 0], No, not-an-inverse-function).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-swapped-relation-audit · candidate

28. Converse Relation before Inverse-Function Qualification

Use the complete total map Fin(1)={0,…,0}→Fin(1) with images=[0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (sorted swapped relation pairs [f(x),x], each codomain fiber size, inverse-function eligibility, inverse table or not-an-inverse-function).

View answer and explanation

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

Swapping pairs creates a relation; it creates a total inverse function exactly when every target has one preimage. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 0]], [1], Yes, [0]).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-swapped-relation-audit · candidate

29. Converse Relation before Inverse-Function Qualification

Use the complete total map Fin(3)={0,…,2}→Fin(3) with images=[0, 1, 2] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (sorted swapped relation pairs [f(x),x], each codomain fiber size, inverse-function eligibility, inverse table or not-an-inverse-function).

View answer and explanation

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

Swapping pairs creates a relation; it creates a total inverse function exactly when every target has one preimage. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 0], [1, 1], [2, 2]], [1, 1, 1], Yes, [0, 1, 2]).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-swapped-relation-audit · candidate

30. Converse Relation before Inverse-Function Qualification

Use the complete total map Fin(3)={0,…,2}→Fin(3) with images=[0, 0, 1] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (sorted swapped relation pairs [f(x),x], each codomain fiber size, inverse-function eligibility, inverse table or not-an-inverse-function).

View answer and explanation

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

Swapping pairs creates a relation; it creates a total inverse function exactly when every target has one preimage. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 0], [0, 1], [1, 2]], [2, 1, 0], No, not-an-inverse-function).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-swapped-relation-audit · candidate

31. Converse Relation before Inverse-Function Qualification

Use the complete total map Fin(3)={0,…,2}→Fin(2) with images=[0, 1, 1] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (sorted swapped relation pairs [f(x),x], each codomain fiber size, inverse-function eligibility, inverse table or not-an-inverse-function).

View answer and explanation

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

Swapping pairs creates a relation; it creates a total inverse function exactly when every target has one preimage. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 0], [1, 1], [1, 2]], [1, 2], No, not-an-inverse-function).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-swapped-relation-audit · candidate

32. Converse Relation before Inverse-Function Qualification

Use the complete total map Fin(2)={0,…,1}→Fin(3) with images=[0, 1] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (sorted swapped relation pairs [f(x),x], each codomain fiber size, inverse-function eligibility, inverse table or not-an-inverse-function).

View answer and explanation

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

Swapping pairs creates a relation; it creates a total inverse function exactly when every target has one preimage. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 0], [1, 1]], [1, 1, 0], No, not-an-inverse-function).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-swapped-relation-audit · candidate

33. Converse Relation before Inverse-Function Qualification

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 0, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (sorted swapped relation pairs [f(x),x], each codomain fiber size, inverse-function eligibility, inverse table or not-an-inverse-function).

View answer and explanation

([[0, 0], [0, 1], [0, 2], [0, 3], [0, 4]], [5, 0, 0], No, not-an-inverse-function)

Swapping pairs creates a relation; it creates a total inverse function exactly when every target has one preimage. 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, 3], [0, 4]], [5, 0, 0], No, not-an-inverse-function).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-swapped-relation-audit · candidate

34. Converse Relation before Inverse-Function Qualification

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 0, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (sorted swapped relation pairs [f(x),x], each codomain fiber size, inverse-function eligibility, inverse table or not-an-inverse-function).

View answer and explanation

([[0, 1], [0, 2], [0, 3], [0, 4], [1, 0]], [4, 1, 0], No, not-an-inverse-function)

Swapping pairs creates a relation; it creates a total inverse function exactly when every target has one preimage. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [0, 2], [0, 3], [0, 4], [1, 0]], [4, 1, 0], No, not-an-inverse-function).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-swapped-relation-audit · candidate

35. Converse Relation before Inverse-Function Qualification

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 0, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (sorted swapped relation pairs [f(x),x], each codomain fiber size, inverse-function eligibility, inverse table or not-an-inverse-function).

View answer and explanation

([[0, 1], [0, 2], [0, 3], [0, 4], [2, 0]], [4, 0, 1], No, not-an-inverse-function)

Swapping pairs creates a relation; it creates a total inverse function exactly when every target has one preimage. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [0, 2], [0, 3], [0, 4], [2, 0]], [4, 0, 1], No, not-an-inverse-function).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-swapped-relation-audit · candidate

36. Converse Relation before Inverse-Function Qualification

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 1, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (sorted swapped relation pairs [f(x),x], each codomain fiber size, inverse-function eligibility, inverse table or not-an-inverse-function).

View answer and explanation

([[0, 0], [0, 2], [0, 3], [0, 4], [1, 1]], [4, 1, 0], No, not-an-inverse-function)

Swapping pairs creates a relation; it creates a total inverse function exactly when every target has one preimage. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 0], [0, 2], [0, 3], [0, 4], [1, 1]], [4, 1, 0], No, not-an-inverse-function).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-swapped-relation-audit · candidate

37. Converse Relation before Inverse-Function Qualification

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 1, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (sorted swapped relation pairs [f(x),x], each codomain fiber size, inverse-function eligibility, inverse table or not-an-inverse-function).

View answer and explanation

([[0, 2], [0, 3], [0, 4], [1, 0], [1, 1]], [3, 2, 0], No, not-an-inverse-function)

Swapping pairs creates a relation; it creates a total inverse function exactly when every target has one preimage. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 2], [0, 3], [0, 4], [1, 0], [1, 1]], [3, 2, 0], No, not-an-inverse-function).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-swapped-relation-audit · candidate

38. Converse Relation before Inverse-Function Qualification

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 1, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (sorted swapped relation pairs [f(x),x], each codomain fiber size, inverse-function eligibility, inverse table or not-an-inverse-function).

View answer and explanation

([[0, 2], [0, 3], [0, 4], [1, 1], [2, 0]], [3, 1, 1], No, not-an-inverse-function)

Swapping pairs creates a relation; it creates a total inverse function exactly when every target has one preimage. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 2], [0, 3], [0, 4], [1, 1], [2, 0]], [3, 1, 1], No, not-an-inverse-function).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-swapped-relation-audit · candidate

39. Converse Relation before Inverse-Function Qualification

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 2, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (sorted swapped relation pairs [f(x),x], each codomain fiber size, inverse-function eligibility, inverse table or not-an-inverse-function).

View answer and explanation

([[0, 0], [0, 2], [0, 3], [0, 4], [2, 1]], [4, 0, 1], No, not-an-inverse-function)

Swapping pairs creates a relation; it creates a total inverse function exactly when every target has one preimage. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 0], [0, 2], [0, 3], [0, 4], [2, 1]], [4, 0, 1], No, not-an-inverse-function).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-swapped-relation-audit · candidate

40. Converse Relation before Inverse-Function Qualification

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 2, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (sorted swapped relation pairs [f(x),x], each codomain fiber size, inverse-function eligibility, inverse table or not-an-inverse-function).

View answer and explanation

([[0, 2], [0, 3], [0, 4], [1, 0], [2, 1]], [3, 1, 1], No, not-an-inverse-function)

Swapping pairs creates a relation; it creates a total inverse function exactly when every target has one preimage. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 2], [0, 3], [0, 4], [1, 0], [2, 1]], [3, 1, 1], No, not-an-inverse-function).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-swapped-relation-audit · candidate

41. Converse Relation before Inverse-Function Qualification

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 2, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (sorted swapped relation pairs [f(x),x], each codomain fiber size, inverse-function eligibility, inverse table or not-an-inverse-function).

View answer and explanation

([[0, 2], [0, 3], [0, 4], [2, 0], [2, 1]], [3, 0, 2], No, not-an-inverse-function)

Swapping pairs creates a relation; it creates a total inverse function exactly when every target has one preimage. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 2], [0, 3], [0, 4], [2, 0], [2, 1]], [3, 0, 2], No, not-an-inverse-function).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-swapped-relation-audit · candidate

42. Converse Relation before Inverse-Function Qualification

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 0, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (sorted swapped relation pairs [f(x),x], each codomain fiber size, inverse-function eligibility, inverse table or not-an-inverse-function).

View answer and explanation

([[0, 0], [0, 1], [0, 3], [0, 4], [1, 2]], [4, 1, 0], No, not-an-inverse-function)

Swapping pairs creates a relation; it creates a total inverse function exactly when every target has one preimage. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 0], [0, 1], [0, 3], [0, 4], [1, 2]], [4, 1, 0], No, not-an-inverse-function).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-swapped-relation-audit · candidate

43. Converse Relation before Inverse-Function Qualification

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 0, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (sorted swapped relation pairs [f(x),x], each codomain fiber size, inverse-function eligibility, inverse table or not-an-inverse-function).

View answer and explanation

([[0, 1], [0, 3], [0, 4], [1, 0], [1, 2]], [3, 2, 0], No, not-an-inverse-function)

Swapping pairs creates a relation; it creates a total inverse function exactly when every target has one preimage. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [0, 3], [0, 4], [1, 0], [1, 2]], [3, 2, 0], No, not-an-inverse-function).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-swapped-relation-audit · candidate

44. Converse Relation before Inverse-Function Qualification

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 0, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (sorted swapped relation pairs [f(x),x], each codomain fiber size, inverse-function eligibility, inverse table or not-an-inverse-function).

View answer and explanation

([[0, 1], [0, 3], [0, 4], [1, 2], [2, 0]], [3, 1, 1], No, not-an-inverse-function)

Swapping pairs creates a relation; it creates a total inverse function exactly when every target has one preimage. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 1], [0, 3], [0, 4], [1, 2], [2, 0]], [3, 1, 1], No, not-an-inverse-function).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-swapped-relation-audit · candidate

45. Converse Relation before Inverse-Function Qualification

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 1, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (sorted swapped relation pairs [f(x),x], each codomain fiber size, inverse-function eligibility, inverse table or not-an-inverse-function).

View answer and explanation

([[0, 0], [0, 3], [0, 4], [1, 1], [1, 2]], [3, 2, 0], No, not-an-inverse-function)

Swapping pairs creates a relation; it creates a total inverse function exactly when every target has one preimage. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 0], [0, 3], [0, 4], [1, 1], [1, 2]], [3, 2, 0], No, not-an-inverse-function).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-swapped-relation-audit · candidate

46. Converse Relation before Inverse-Function Qualification

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 1, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (sorted swapped relation pairs [f(x),x], each codomain fiber size, inverse-function eligibility, inverse table or not-an-inverse-function).

View answer and explanation

([[0, 3], [0, 4], [1, 0], [1, 1], [1, 2]], [2, 3, 0], No, not-an-inverse-function)

Swapping pairs creates a relation; it creates a total inverse function exactly when every target has one preimage. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 3], [0, 4], [1, 0], [1, 1], [1, 2]], [2, 3, 0], No, not-an-inverse-function).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-swapped-relation-audit · candidate

47. Converse Relation before Inverse-Function Qualification

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 1, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (sorted swapped relation pairs [f(x),x], each codomain fiber size, inverse-function eligibility, inverse table or not-an-inverse-function).

View answer and explanation

([[0, 3], [0, 4], [1, 1], [1, 2], [2, 0]], [2, 2, 1], No, not-an-inverse-function)

Swapping pairs creates a relation; it creates a total inverse function exactly when every target has one preimage. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 3], [0, 4], [1, 1], [1, 2], [2, 0]], [2, 2, 1], No, not-an-inverse-function).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-swapped-relation-audit · candidate

48. Converse Relation before Inverse-Function Qualification

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 2, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (sorted swapped relation pairs [f(x),x], each codomain fiber size, inverse-function eligibility, inverse table or not-an-inverse-function).

View answer and explanation

([[0, 0], [0, 3], [0, 4], [1, 2], [2, 1]], [3, 1, 1], No, not-an-inverse-function)

Swapping pairs creates a relation; it creates a total inverse function exactly when every target has one preimage. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 0], [0, 3], [0, 4], [1, 2], [2, 1]], [3, 1, 1], No, not-an-inverse-function).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-swapped-relation-audit · candidate

49. Converse Relation before Inverse-Function Qualification

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 2, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (sorted swapped relation pairs [f(x),x], each codomain fiber size, inverse-function eligibility, inverse table or not-an-inverse-function).

View answer and explanation

([[0, 3], [0, 4], [1, 0], [1, 2], [2, 1]], [2, 2, 1], No, not-an-inverse-function)

Swapping pairs creates a relation; it creates a total inverse function exactly when every target has one preimage. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 3], [0, 4], [1, 0], [1, 2], [2, 1]], [2, 2, 1], No, not-an-inverse-function).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-swapped-relation-audit · candidate

50. Converse Relation before Inverse-Function Qualification

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 2, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (sorted swapped relation pairs [f(x),x], each codomain fiber size, inverse-function eligibility, inverse table or not-an-inverse-function).

View answer and explanation

([[0, 3], [0, 4], [1, 2], [2, 0], [2, 1]], [2, 1, 2], No, not-an-inverse-function)

Swapping pairs creates a relation; it creates a total inverse function exactly when every target has one preimage. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 3], [0, 4], [1, 2], [2, 0], [2, 1]], [2, 1, 2], No, not-an-inverse-function).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-swapped-relation-audit · candidate

51. Swap Two Output Labels and Recheck Bijectivity

Use the complete total map Fin(0)=∅→Fin(2) with images=[] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Interchange codomain labels a=0, b=1 in every image (equal labels are allowed). Give (original images, new images, new fibers, original bijectivity, new bijectivity, do these two flags agree?).

View answer and explanation

([], [], [[], []], No, No, Yes)

Apply the same output-label transposition to every input and recompute the complete fibers. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [], [[], []], No, No, Yes).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-output-transposition · candidate

52. Swap Two Output Labels and Recheck Bijectivity

Use the complete total map Fin(2)={0,…,1}→Fin(2) with images=[0, 1] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Interchange codomain labels a=0, b=1 in every image (equal labels are allowed). Give (original images, new images, new fibers, original bijectivity, new bijectivity, do these two flags agree?).

View answer and explanation

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

Apply the same output-label transposition to every input and recompute the complete fibers. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1], [1, 0], [[1], [0]], Yes, Yes, Yes).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-output-transposition · candidate

53. Swap Two Output Labels and Recheck Bijectivity

Use the complete total map Fin(3)={0,…,2}→Fin(3) with images=[0, 0, 1] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Interchange codomain labels a=0, b=1 in every image (equal labels are allowed). Give (original images, new images, new fibers, original bijectivity, new bijectivity, do these two flags agree?).

View answer and explanation

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

Apply the same output-label transposition to every input and recompute the complete fibers. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1], [1, 1, 0], [[2], [0, 1], []], No, No, Yes).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-output-transposition · candidate

54. Swap Two Output Labels and Recheck Bijectivity

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 0, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Interchange codomain labels a=0, b=2 in every image (equal labels are allowed). Give (original images, new images, new fibers, original bijectivity, new bijectivity, do these two flags agree?).

View answer and explanation

([0, 0, 0, 0, 0], [2, 2, 2, 2, 2], [[], [], [0, 1, 2, 3, 4]], No, No, Yes)

Apply the same output-label transposition to every input and recompute the complete fibers. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 0, 0, 0], [2, 2, 2, 2, 2], [[], [], [0, 1, 2, 3, 4]], No, No, Yes).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-output-transposition · candidate

55. Swap Two Output Labels and Recheck Bijectivity

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 0, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Interchange codomain labels a=0, b=2 in every image (equal labels are allowed). Give (original images, new images, new fibers, original bijectivity, new bijectivity, do these two flags agree?).

View answer and explanation

([1, 0, 0, 0, 0], [1, 2, 2, 2, 2], [[], [0], [1, 2, 3, 4]], No, No, Yes)

Apply the same output-label transposition to every input and recompute the complete fibers. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 0, 0, 0], [1, 2, 2, 2, 2], [[], [0], [1, 2, 3, 4]], No, No, Yes).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-output-transposition · candidate

56. Swap Two Output Labels and Recheck Bijectivity

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 0, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Interchange codomain labels a=0, b=2 in every image (equal labels are allowed). Give (original images, new images, new fibers, original bijectivity, new bijectivity, do these two flags agree?).

View answer and explanation

([2, 0, 0, 0, 0], [0, 2, 2, 2, 2], [[0], [], [1, 2, 3, 4]], No, No, Yes)

Apply the same output-label transposition to every input and recompute the complete fibers. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 0, 0, 0], [0, 2, 2, 2, 2], [[0], [], [1, 2, 3, 4]], No, No, Yes).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-output-transposition · candidate

57. Swap Two Output Labels and Recheck Bijectivity

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 1, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Interchange codomain labels a=0, b=2 in every image (equal labels are allowed). Give (original images, new images, new fibers, original bijectivity, new bijectivity, do these two flags agree?).

View answer and explanation

([0, 1, 0, 0, 0], [2, 1, 2, 2, 2], [[], [1], [0, 2, 3, 4]], No, No, Yes)

Apply the same output-label transposition to every input and recompute the complete fibers. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 0, 0, 0], [2, 1, 2, 2, 2], [[], [1], [0, 2, 3, 4]], No, No, Yes).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-output-transposition · candidate

58. Swap Two Output Labels and Recheck Bijectivity

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 1, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Interchange codomain labels a=0, b=2 in every image (equal labels are allowed). Give (original images, new images, new fibers, original bijectivity, new bijectivity, do these two flags agree?).

View answer and explanation

([1, 1, 0, 0, 0], [1, 1, 2, 2, 2], [[], [0, 1], [2, 3, 4]], No, No, Yes)

Apply the same output-label transposition to every input and recompute the complete fibers. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 0, 0, 0], [1, 1, 2, 2, 2], [[], [0, 1], [2, 3, 4]], No, No, Yes).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-output-transposition · candidate

59. Swap Two Output Labels and Recheck Bijectivity

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 1, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Interchange codomain labels a=0, b=2 in every image (equal labels are allowed). Give (original images, new images, new fibers, original bijectivity, new bijectivity, do these two flags agree?).

View answer and explanation

([2, 1, 0, 0, 0], [0, 1, 2, 2, 2], [[0], [1], [2, 3, 4]], No, No, Yes)

Apply the same output-label transposition to every input and recompute the complete fibers. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 0, 0, 0], [0, 1, 2, 2, 2], [[0], [1], [2, 3, 4]], No, No, Yes).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-output-transposition · candidate

60. Swap Two Output Labels and Recheck Bijectivity

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 2, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Interchange codomain labels a=0, b=2 in every image (equal labels are allowed). Give (original images, new images, new fibers, original bijectivity, new bijectivity, do these two flags agree?).

View answer and explanation

([0, 2, 0, 0, 0], [2, 0, 2, 2, 2], [[1], [], [0, 2, 3, 4]], No, No, Yes)

Apply the same output-label transposition to every input and recompute the complete fibers. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 0, 0, 0], [2, 0, 2, 2, 2], [[1], [], [0, 2, 3, 4]], No, No, Yes).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-output-transposition · candidate

61. Swap Two Output Labels and Recheck Bijectivity

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 2, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Interchange codomain labels a=0, b=2 in every image (equal labels are allowed). Give (original images, new images, new fibers, original bijectivity, new bijectivity, do these two flags agree?).

View answer and explanation

([1, 2, 0, 0, 0], [1, 0, 2, 2, 2], [[1], [0], [2, 3, 4]], No, No, Yes)

Apply the same output-label transposition to every input and recompute the complete fibers. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 0, 0, 0], [1, 0, 2, 2, 2], [[1], [0], [2, 3, 4]], No, No, Yes).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-output-transposition · candidate

62. Swap Two Output Labels and Recheck Bijectivity

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 2, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Interchange codomain labels a=0, b=2 in every image (equal labels are allowed). Give (original images, new images, new fibers, original bijectivity, new bijectivity, do these two flags agree?).

View answer and explanation

([2, 2, 0, 0, 0], [0, 0, 2, 2, 2], [[0, 1], [], [2, 3, 4]], No, No, Yes)

Apply the same output-label transposition to every input and recompute the complete fibers. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2, 0, 0, 0], [0, 0, 2, 2, 2], [[0, 1], [], [2, 3, 4]], No, No, Yes).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-output-transposition · candidate

63. Swap Two Output Labels and Recheck Bijectivity

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 0, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Interchange codomain labels a=0, b=2 in every image (equal labels are allowed). Give (original images, new images, new fibers, original bijectivity, new bijectivity, do these two flags agree?).

View answer and explanation

([0, 0, 1, 0, 0], [2, 2, 1, 2, 2], [[], [2], [0, 1, 3, 4]], No, No, Yes)

Apply the same output-label transposition to every input and recompute the complete fibers. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1, 0, 0], [2, 2, 1, 2, 2], [[], [2], [0, 1, 3, 4]], No, No, Yes).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-output-transposition · candidate

64. Swap Two Output Labels and Recheck Bijectivity

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 0, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Interchange codomain labels a=0, b=2 in every image (equal labels are allowed). Give (original images, new images, new fibers, original bijectivity, new bijectivity, do these two flags agree?).

View answer and explanation

([1, 0, 1, 0, 0], [1, 2, 1, 2, 2], [[], [0, 2], [1, 3, 4]], No, No, Yes)

Apply the same output-label transposition to every input and recompute the complete fibers. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 1, 0, 0], [1, 2, 1, 2, 2], [[], [0, 2], [1, 3, 4]], No, No, Yes).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-output-transposition · candidate

65. Swap Two Output Labels and Recheck Bijectivity

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 0, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Interchange codomain labels a=0, b=2 in every image (equal labels are allowed). Give (original images, new images, new fibers, original bijectivity, new bijectivity, do these two flags agree?).

View answer and explanation

([2, 0, 1, 0, 0], [0, 2, 1, 2, 2], [[0], [2], [1, 3, 4]], No, No, Yes)

Apply the same output-label transposition to every input and recompute the complete fibers. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 1, 0, 0], [0, 2, 1, 2, 2], [[0], [2], [1, 3, 4]], No, No, Yes).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-output-transposition · candidate

66. Swap Two Output Labels and Recheck Bijectivity

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 1, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Interchange codomain labels a=0, b=2 in every image (equal labels are allowed). Give (original images, new images, new fibers, original bijectivity, new bijectivity, do these two flags agree?).

View answer and explanation

([0, 1, 1, 0, 0], [2, 1, 1, 2, 2], [[], [1, 2], [0, 3, 4]], No, No, Yes)

Apply the same output-label transposition to every input and recompute the complete fibers. 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, 1, 2, 2], [[], [1, 2], [0, 3, 4]], No, No, Yes).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-output-transposition · candidate

67. Swap Two Output Labels and Recheck Bijectivity

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 1, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Interchange codomain labels a=0, b=2 in every image (equal labels are allowed). Give (original images, new images, new fibers, original bijectivity, new bijectivity, do these two flags agree?).

View answer and explanation

([1, 1, 1, 0, 0], [1, 1, 1, 2, 2], [[], [0, 1, 2], [3, 4]], No, No, Yes)

Apply the same output-label transposition to every input and recompute the complete fibers. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 1, 0, 0], [1, 1, 1, 2, 2], [[], [0, 1, 2], [3, 4]], No, No, Yes).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-output-transposition · candidate

68. Swap Two Output Labels and Recheck Bijectivity

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 1, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Interchange codomain labels a=0, b=2 in every image (equal labels are allowed). Give (original images, new images, new fibers, original bijectivity, new bijectivity, do these two flags agree?).

View answer and explanation

([2, 1, 1, 0, 0], [0, 1, 1, 2, 2], [[0], [1, 2], [3, 4]], No, No, Yes)

Apply the same output-label transposition to every input and recompute the complete fibers. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 1, 0, 0], [0, 1, 1, 2, 2], [[0], [1, 2], [3, 4]], No, No, Yes).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-output-transposition · candidate

69. Swap Two Output Labels and Recheck Bijectivity

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 2, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Interchange codomain labels a=0, b=2 in every image (equal labels are allowed). Give (original images, new images, new fibers, original bijectivity, new bijectivity, do these two flags agree?).

View answer and explanation

([0, 2, 1, 0, 0], [2, 0, 1, 2, 2], [[1], [2], [0, 3, 4]], No, No, Yes)

Apply the same output-label transposition to every input and recompute the complete fibers. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 1, 0, 0], [2, 0, 1, 2, 2], [[1], [2], [0, 3, 4]], No, No, Yes).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-output-transposition · candidate

70. Swap Two Output Labels and Recheck Bijectivity

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 2, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Interchange codomain labels a=0, b=2 in every image (equal labels are allowed). Give (original images, new images, new fibers, original bijectivity, new bijectivity, do these two flags agree?).

View answer and explanation

([1, 2, 1, 0, 0], [1, 0, 1, 2, 2], [[1], [0, 2], [3, 4]], No, No, Yes)

Apply the same output-label transposition to every input and recompute the complete fibers. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 1, 0, 0], [1, 0, 1, 2, 2], [[1], [0, 2], [3, 4]], No, No, Yes).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-output-transposition · candidate

71. Swap Two Output Labels and Recheck Bijectivity

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 2, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Interchange codomain labels a=0, b=2 in every image (equal labels are allowed). Give (original images, new images, new fibers, original bijectivity, new bijectivity, do these two flags agree?).

View answer and explanation

([2, 2, 1, 0, 0], [0, 0, 1, 2, 2], [[0, 1], [2], [3, 4]], No, No, Yes)

Apply the same output-label transposition to every input and recompute the complete fibers. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2, 1, 0, 0], [0, 0, 1, 2, 2], [[0, 1], [2], [3, 4]], No, No, Yes).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-output-transposition · candidate

72. Swap Two Output Labels and Recheck Bijectivity

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 0, 2, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Interchange codomain labels a=0, b=2 in every image (equal labels are allowed). Give (original images, new images, new fibers, original bijectivity, new bijectivity, do these two flags agree?).

View answer and explanation

([0, 0, 2, 0, 0], [2, 2, 0, 2, 2], [[2], [], [0, 1, 3, 4]], No, No, Yes)

Apply the same output-label transposition to every input and recompute the complete fibers. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 2, 0, 0], [2, 2, 0, 2, 2], [[2], [], [0, 1, 3, 4]], No, No, Yes).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-output-transposition · candidate

73. Swap Two Output Labels and Recheck Bijectivity

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 0, 2, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Interchange codomain labels a=0, b=2 in every image (equal labels are allowed). Give (original images, new images, new fibers, original bijectivity, new bijectivity, do these two flags agree?).

View answer and explanation

([1, 0, 2, 0, 0], [1, 2, 0, 2, 2], [[2], [0], [1, 3, 4]], No, No, Yes)

Apply the same output-label transposition to every input and recompute the complete fibers. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 2, 0, 0], [1, 2, 0, 2, 2], [[2], [0], [1, 3, 4]], No, No, Yes).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-output-transposition · candidate

74. Swap Two Output Labels and Recheck Bijectivity

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 0, 2, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Interchange codomain labels a=0, b=2 in every image (equal labels are allowed). Give (original images, new images, new fibers, original bijectivity, new bijectivity, do these two flags agree?).

View answer and explanation

([2, 0, 2, 0, 0], [0, 2, 0, 2, 2], [[0, 2], [], [1, 3, 4]], No, No, Yes)

Apply the same output-label transposition to every input and recompute the complete fibers. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 2, 0, 0], [0, 2, 0, 2, 2], [[0, 2], [], [1, 3, 4]], No, No, Yes).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-output-transposition · candidate

75. Swap Two Output Labels and Recheck Bijectivity

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 1, 2, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Interchange codomain labels a=0, b=2 in every image (equal labels are allowed). Give (original images, new images, new fibers, original bijectivity, new bijectivity, do these two flags agree?).

View answer and explanation

([0, 1, 2, 0, 0], [2, 1, 0, 2, 2], [[2], [1], [0, 3, 4]], No, No, Yes)

Apply the same output-label transposition to every input and recompute the complete fibers. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2, 0, 0], [2, 1, 0, 2, 2], [[2], [1], [0, 3, 4]], No, No, Yes).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-output-transposition · candidate

76. Equal Cardinality versus the Given Total Table

Use the complete total map Fin(0)=∅→Fin(0) with images=[] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give ([n,m], images, n=m?, every fiber, is the given map bijective?).

View answer and explanation

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

Equal finite sizes address cardinality, not whether this particular table has repeated or missing values. 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

Bijectivity and unique preimages · qf-density17-bij-size-obstruction-with-table · candidate

77. Equal Cardinality versus the Given Total Table

Use the complete total map Fin(0)=∅→Fin(2) with images=[] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give ([n,m], images, n=m?, every fiber, is the given map bijective?).

View answer and explanation

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

Equal finite sizes address cardinality, not whether this particular table has repeated or missing values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2], [], No, [[], []], No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-size-obstruction-with-table · candidate

78. Equal Cardinality versus the Given Total Table

Use the complete total map Fin(1)={0,…,0}→Fin(1) with images=[0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give ([n,m], images, n=m?, every fiber, is the given map bijective?).

View answer and explanation

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

Equal finite sizes address cardinality, not whether this particular table has repeated or missing values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1], [0], Yes, [[0]], Yes).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-size-obstruction-with-table · candidate

79. Equal Cardinality versus the Given Total Table

Use the complete total map Fin(3)={0,…,2}→Fin(3) with images=[0, 1, 2] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give ([n,m], images, n=m?, every fiber, is the given map bijective?).

View answer and explanation

([3, 3], [0, 1, 2], Yes, [[0], [1], [2]], Yes)

Equal finite sizes address cardinality, not whether this particular table has repeated or missing values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [0, 1, 2], Yes, [[0], [1], [2]], Yes).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-size-obstruction-with-table · candidate

80. Equal Cardinality versus the Given Total Table

Use the complete total map Fin(3)={0,…,2}→Fin(3) with images=[0, 0, 1] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give ([n,m], images, n=m?, every fiber, is the given map bijective?).

View answer and explanation

([3, 3], [0, 0, 1], Yes, [[0, 1], [2], []], No)

Equal finite sizes address cardinality, not whether this particular table has repeated or missing values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [0, 0, 1], Yes, [[0, 1], [2], []], No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-size-obstruction-with-table · candidate

81. Equal Cardinality versus the Given Total Table

Use the complete total map Fin(3)={0,…,2}→Fin(2) with images=[0, 1, 1] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give ([n,m], images, n=m?, every fiber, is the given map bijective?).

View answer and explanation

([3, 2], [0, 1, 1], No, [[0], [1, 2]], No)

Equal finite sizes address cardinality, not whether this particular table has repeated or missing values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 2], [0, 1, 1], No, [[0], [1, 2]], No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-size-obstruction-with-table · candidate

82. Equal Cardinality versus the Given Total Table

Use the complete total map Fin(2)={0,…,1}→Fin(3) with images=[0, 1] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give ([n,m], images, n=m?, every fiber, is the given map bijective?).

View answer and explanation

([2, 3], [0, 1], No, [[0], [1], []], No)

Equal finite sizes address cardinality, not whether this particular table has repeated or missing values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 3], [0, 1], No, [[0], [1], []], No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-size-obstruction-with-table · candidate

83. Equal Cardinality versus the Given Total Table

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 0, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give ([n,m], images, n=m?, every fiber, is the given map bijective?).

View answer and explanation

([5, 3], [0, 0, 0, 0, 0], No, [[0, 1, 2, 3, 4], [], []], No)

Equal finite sizes address cardinality, not whether this particular table has repeated or missing values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 3], [0, 0, 0, 0, 0], No, [[0, 1, 2, 3, 4], [], []], No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-size-obstruction-with-table · candidate

84. Equal Cardinality versus the Given Total Table

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 0, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give ([n,m], images, n=m?, every fiber, is the given map bijective?).

View answer and explanation

([5, 3], [1, 0, 0, 0, 0], No, [[1, 2, 3, 4], [0], []], No)

Equal finite sizes address cardinality, not whether this particular table has repeated or missing values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 3], [1, 0, 0, 0, 0], No, [[1, 2, 3, 4], [0], []], No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-size-obstruction-with-table · candidate

85. Equal Cardinality versus the Given Total Table

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 0, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give ([n,m], images, n=m?, every fiber, is the given map bijective?).

View answer and explanation

([5, 3], [2, 0, 0, 0, 0], No, [[1, 2, 3, 4], [], [0]], No)

Equal finite sizes address cardinality, not whether this particular table has repeated or missing values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 3], [2, 0, 0, 0, 0], No, [[1, 2, 3, 4], [], [0]], No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-size-obstruction-with-table · candidate

86. Equal Cardinality versus the Given Total Table

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 1, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give ([n,m], images, n=m?, every fiber, is the given map bijective?).

View answer and explanation

([5, 3], [0, 1, 0, 0, 0], No, [[0, 2, 3, 4], [1], []], No)

Equal finite sizes address cardinality, not whether this particular table has repeated or missing values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 3], [0, 1, 0, 0, 0], No, [[0, 2, 3, 4], [1], []], No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-size-obstruction-with-table · candidate

87. Equal Cardinality versus the Given Total Table

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 1, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give ([n,m], images, n=m?, every fiber, is the given map bijective?).

View answer and explanation

([5, 3], [1, 1, 0, 0, 0], No, [[2, 3, 4], [0, 1], []], No)

Equal finite sizes address cardinality, not whether this particular table has repeated or missing values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 3], [1, 1, 0, 0, 0], No, [[2, 3, 4], [0, 1], []], No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-size-obstruction-with-table · candidate

88. Equal Cardinality versus the Given Total Table

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 1, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give ([n,m], images, n=m?, every fiber, is the given map bijective?).

View answer and explanation

([5, 3], [2, 1, 0, 0, 0], No, [[2, 3, 4], [1], [0]], No)

Equal finite sizes address cardinality, not whether this particular table has repeated or missing values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 3], [2, 1, 0, 0, 0], No, [[2, 3, 4], [1], [0]], No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-size-obstruction-with-table · candidate

89. Equal Cardinality versus the Given Total Table

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 2, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give ([n,m], images, n=m?, every fiber, is the given map bijective?).

View answer and explanation

([5, 3], [0, 2, 0, 0, 0], No, [[0, 2, 3, 4], [], [1]], No)

Equal finite sizes address cardinality, not whether this particular table has repeated or missing values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 3], [0, 2, 0, 0, 0], No, [[0, 2, 3, 4], [], [1]], No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-size-obstruction-with-table · candidate

90. Equal Cardinality versus the Given Total Table

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 2, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give ([n,m], images, n=m?, every fiber, is the given map bijective?).

View answer and explanation

([5, 3], [1, 2, 0, 0, 0], No, [[2, 3, 4], [0], [1]], No)

Equal finite sizes address cardinality, not whether this particular table has repeated or missing values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 3], [1, 2, 0, 0, 0], No, [[2, 3, 4], [0], [1]], No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-size-obstruction-with-table · candidate

91. Equal Cardinality versus the Given Total Table

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 2, 0, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give ([n,m], images, n=m?, every fiber, is the given map bijective?).

View answer and explanation

([5, 3], [2, 2, 0, 0, 0], No, [[2, 3, 4], [], [0, 1]], No)

Equal finite sizes address cardinality, not whether this particular table has repeated or missing values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 3], [2, 2, 0, 0, 0], No, [[2, 3, 4], [], [0, 1]], No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-size-obstruction-with-table · candidate

92. Equal Cardinality versus the Given Total Table

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 0, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give ([n,m], images, n=m?, every fiber, is the given map bijective?).

View answer and explanation

([5, 3], [0, 0, 1, 0, 0], No, [[0, 1, 3, 4], [2], []], No)

Equal finite sizes address cardinality, not whether this particular table has repeated or missing values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 3], [0, 0, 1, 0, 0], No, [[0, 1, 3, 4], [2], []], No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-size-obstruction-with-table · candidate

93. Equal Cardinality versus the Given Total Table

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 0, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give ([n,m], images, n=m?, every fiber, is the given map bijective?).

View answer and explanation

([5, 3], [1, 0, 1, 0, 0], No, [[1, 3, 4], [0, 2], []], No)

Equal finite sizes address cardinality, not whether this particular table has repeated or missing values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 3], [1, 0, 1, 0, 0], No, [[1, 3, 4], [0, 2], []], No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-size-obstruction-with-table · candidate

94. Equal Cardinality versus the Given Total Table

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 0, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give ([n,m], images, n=m?, every fiber, is the given map bijective?).

View answer and explanation

([5, 3], [2, 0, 1, 0, 0], No, [[1, 3, 4], [2], [0]], No)

Equal finite sizes address cardinality, not whether this particular table has repeated or missing values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 3], [2, 0, 1, 0, 0], No, [[1, 3, 4], [2], [0]], No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-size-obstruction-with-table · candidate

95. Equal Cardinality versus the Given Total Table

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 1, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give ([n,m], images, n=m?, every fiber, is the given map bijective?).

View answer and explanation

([5, 3], [0, 1, 1, 0, 0], No, [[0, 3, 4], [1, 2], []], No)

Equal finite sizes address cardinality, not whether this particular table has repeated or missing values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 3], [0, 1, 1, 0, 0], No, [[0, 3, 4], [1, 2], []], No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-size-obstruction-with-table · candidate

96. Equal Cardinality versus the Given Total Table

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 1, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give ([n,m], images, n=m?, every fiber, is the given map bijective?).

View answer and explanation

([5, 3], [1, 1, 1, 0, 0], No, [[3, 4], [0, 1, 2], []], No)

Equal finite sizes address cardinality, not whether this particular table has repeated or missing values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 3], [1, 1, 1, 0, 0], No, [[3, 4], [0, 1, 2], []], No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-size-obstruction-with-table · candidate

97. Equal Cardinality versus the Given Total Table

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 1, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give ([n,m], images, n=m?, every fiber, is the given map bijective?).

View answer and explanation

([5, 3], [2, 1, 1, 0, 0], No, [[3, 4], [1, 2], [0]], No)

Equal finite sizes address cardinality, not whether this particular table has repeated or missing values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 3], [2, 1, 1, 0, 0], No, [[3, 4], [1, 2], [0]], No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-size-obstruction-with-table · candidate

98. Equal Cardinality versus the Given Total Table

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 2, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give ([n,m], images, n=m?, every fiber, is the given map bijective?).

View answer and explanation

([5, 3], [0, 2, 1, 0, 0], No, [[0, 3, 4], [2], [1]], No)

Equal finite sizes address cardinality, not whether this particular table has repeated or missing values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 3], [0, 2, 1, 0, 0], No, [[0, 3, 4], [2], [1]], No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-size-obstruction-with-table · candidate

99. Equal Cardinality versus the Given Total Table

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 2, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give ([n,m], images, n=m?, every fiber, is the given map bijective?).

View answer and explanation

([5, 3], [1, 2, 1, 0, 0], No, [[3, 4], [0, 2], [1]], No)

Equal finite sizes address cardinality, not whether this particular table has repeated or missing values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 3], [1, 2, 1, 0, 0], No, [[3, 4], [0, 2], [1]], No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-size-obstruction-with-table · candidate

100. Equal Cardinality versus the Given Total Table

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 2, 1, 0, 0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give ([n,m], images, n=m?, every fiber, is the given map bijective?).

View answer and explanation

([5, 3], [2, 2, 1, 0, 0], No, [[3, 4], [2], [0, 1]], No)

Equal finite sizes address cardinality, not whether this particular table has repeated or missing values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 3], [2, 2, 1, 0, 0], No, [[3, 4], [2], [0, 1]], No).

Reviewed step-by-step derivation

Bijectivity and unique preimages · qf-density17-bij-size-obstruction-with-table · candidate

Canonical question JSONL · English JSONL · Validation report