# Complete Finite-Map Bijections and Cardinality Guards

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

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

batch-20261012-advance-20261017-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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0001/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0002/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0003/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0004/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0005/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0006/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0007/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0008/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0009/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0010/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0011/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0012/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0013/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0014/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0015/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0016/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0017/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0018/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0019/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0020/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0021/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0022/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0023/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0024/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0025/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0026/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0027/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0028/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0029/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0030/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0031/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0032/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0033/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0034/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0035/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0036/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0037/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0038/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0039/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0040/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0041/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0042/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0043/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0044/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0045/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0046/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0047/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0048/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0049/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0050/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0051/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0052/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0053/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0054/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0055/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0056/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0057/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0058/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0059/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0060/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0061/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0062/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0063/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0064/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0065/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0066/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0067/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0068/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0069/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0070/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0071/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0072/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0073/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0074/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0075/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0076/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0077/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0078/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0079/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0080/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0081/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0082/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0083/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0084/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0085/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0086/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0087/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0088/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0089/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0090/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0091/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0092/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0093/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0094/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0095/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0096/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0097/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0098/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0099/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r01-bijective-0100/)

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



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261012-advance-20261017-r01-bijective-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261012-advance-20261017-r01-bijective-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261012-advance-20261017-r01-bijective-foundations/validation-report.json)