OCMEBilingual mathematics & derivations繁體中文

Declared Codomain Fibers, Changes, and Target Guards

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

batch-20261012-advance-20261016-r07-surjective-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. Every Declared Codomain Fiber and Missing Target

Finite types start at 0: Fin(0)=∅→Fin(0), complete images=[]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.

View answer and explanation

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

List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [], [], Yes).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-fiber-inventory · candidate

2. Every Declared Codomain Fiber and Missing Target

Finite types start at 0: Fin(0)=∅→Fin(2), complete images=[]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.

View answer and explanation

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

List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [[], []], [0, 1], No).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-fiber-inventory · candidate

3. Every Declared Codomain Fiber and Missing Target

Finite types start at 0: Fin(3)={0,…,2}→Fin(2), complete images=[0, 1, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.

View answer and explanation

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

List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 1], [[0], [1, 2]], [], Yes).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-fiber-inventory · candidate

4. Every Declared Codomain Fiber and Missing Target

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.

View answer and explanation

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

List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 0, 0, 1], [[0, 1, 2, 3], [4], []], [2], No).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-fiber-inventory · candidate

5. Every Declared Codomain Fiber and Missing Target

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.

View answer and explanation

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

List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 0, 0, 1], [[1, 2, 3], [0, 4], []], [2], No).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-fiber-inventory · candidate

6. Every Declared Codomain Fiber and Missing Target

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.

View answer and explanation

([2, 0, 0, 0, 1], [[1, 2, 3], [4], [0]], [], Yes)

List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 0, 0, 1], [[1, 2, 3], [4], [0]], [], Yes).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-fiber-inventory · candidate

7. Every Declared Codomain Fiber and Missing Target

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.

View answer and explanation

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

List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 0, 0, 1], [[0, 2, 3], [1, 4], []], [2], No).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-fiber-inventory · candidate

8. Every Declared Codomain Fiber and Missing Target

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.

View answer and explanation

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

List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 0, 0, 1], [[2, 3], [0, 1, 4], []], [2], No).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-fiber-inventory · candidate

9. Every Declared Codomain Fiber and Missing Target

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.

View answer and explanation

([2, 1, 0, 0, 1], [[2, 3], [1, 4], [0]], [], Yes)

List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 0, 0, 1], [[2, 3], [1, 4], [0]], [], Yes).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-fiber-inventory · candidate

10. Every Declared Codomain Fiber and Missing Target

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 2, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.

View answer and explanation

([0, 2, 0, 0, 1], [[0, 2, 3], [4], [1]], [], Yes)

List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 0, 0, 1], [[0, 2, 3], [4], [1]], [], Yes).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-fiber-inventory · candidate

11. Every Declared Codomain Fiber and Missing Target

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 2, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.

View answer and explanation

([1, 2, 0, 0, 1], [[2, 3], [0, 4], [1]], [], Yes)

List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 0, 0, 1], [[2, 3], [0, 4], [1]], [], Yes).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-fiber-inventory · candidate

12. Every Declared Codomain Fiber and Missing Target

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 2, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.

View answer and explanation

([2, 2, 0, 0, 1], [[2, 3], [4], [0, 1]], [], Yes)

List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2, 0, 0, 1], [[2, 3], [4], [0, 1]], [], Yes).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-fiber-inventory · candidate

13. Every Declared Codomain Fiber and Missing Target

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.

View answer and explanation

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

List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1, 0, 1], [[0, 1, 3], [2, 4], []], [2], No).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-fiber-inventory · candidate

14. Every Declared Codomain Fiber and Missing Target

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.

View answer and explanation

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

List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 1, 0, 1], [[1, 3], [0, 2, 4], []], [2], No).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-fiber-inventory · candidate

15. Every Declared Codomain Fiber and Missing Target

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 0, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.

View answer and explanation

([2, 0, 1, 0, 1], [[1, 3], [2, 4], [0]], [], Yes)

List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 1, 0, 1], [[1, 3], [2, 4], [0]], [], Yes).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-fiber-inventory · candidate

16. Every Declared Codomain Fiber and Missing Target

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 1, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.

View answer and explanation

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

List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 1, 0, 1], [[0, 3], [1, 2, 4], []], [2], No).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-fiber-inventory · candidate

17. Every Declared Codomain Fiber and Missing Target

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 1, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.

View answer and explanation

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

List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 1, 0, 1], [[3], [0, 1, 2, 4], []], [2], No).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-fiber-inventory · candidate

18. Every Declared Codomain Fiber and Missing Target

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 1, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.

View answer and explanation

([2, 1, 1, 0, 1], [[3], [1, 2, 4], [0]], [], Yes)

List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 1, 0, 1], [[3], [1, 2, 4], [0]], [], Yes).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-fiber-inventory · candidate

19. Every Declared Codomain Fiber and Missing Target

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 2, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.

View answer and explanation

([0, 2, 1, 0, 1], [[0, 3], [2, 4], [1]], [], Yes)

List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 1, 0, 1], [[0, 3], [2, 4], [1]], [], Yes).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-fiber-inventory · candidate

20. Every Declared Codomain Fiber and Missing Target

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 2, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.

View answer and explanation

([1, 2, 1, 0, 1], [[3], [0, 2, 4], [1]], [], Yes)

List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 1, 0, 1], [[3], [0, 2, 4], [1]], [], Yes).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-fiber-inventory · candidate

21. Every Declared Codomain Fiber and Missing Target

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 2, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.

View answer and explanation

([2, 2, 1, 0, 1], [[3], [2, 4], [0, 1]], [], Yes)

List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2, 1, 0, 1], [[3], [2, 4], [0, 1]], [], Yes).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-fiber-inventory · candidate

22. Every Declared Codomain Fiber and Missing Target

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 2, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.

View answer and explanation

([0, 0, 2, 0, 1], [[0, 1, 3], [4], [2]], [], Yes)

List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 2, 0, 1], [[0, 1, 3], [4], [2]], [], Yes).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-fiber-inventory · candidate

23. Every Declared Codomain Fiber and Missing Target

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 2, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.

View answer and explanation

([1, 0, 2, 0, 1], [[1, 3], [0, 4], [2]], [], Yes)

List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 2, 0, 1], [[1, 3], [0, 4], [2]], [], Yes).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-fiber-inventory · candidate

24. Every Declared Codomain Fiber and Missing Target

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 0, 2, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.

View answer and explanation

([2, 0, 2, 0, 1], [[1, 3], [4], [0, 2]], [], Yes)

List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 2, 0, 1], [[1, 3], [4], [0, 2]], [], Yes).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-fiber-inventory · candidate

25. Every Declared Codomain Fiber and Missing Target

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 1, 2, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.

View answer and explanation

([0, 1, 2, 0, 1], [[0, 3], [1, 4], [2]], [], Yes)

List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2, 0, 1], [[0, 3], [1, 4], [2]], [], Yes).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-fiber-inventory · candidate

26. Changed Codomain Must Contain All Original Outputs

Finite types start at 0: Fin(0)=∅→Fin(0), complete images=[]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).

View answer and explanation

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

Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [], Yes, [], Yes).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-changed-codomain-qualification · candidate

27. Changed Codomain Must Contain All Original Outputs

Finite types start at 0: Fin(0)=∅→Fin(1), complete images=[]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[-1] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).

View answer and explanation

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

Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [-1], Yes, [-1], No).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-changed-codomain-qualification · candidate

28. Changed Codomain Must Contain All Original Outputs

Finite types start at 0: Fin(2)={0,…,1}→Fin(3), complete images=[0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).

View answer and explanation

([0, 1], [0], No, outside-codomain, not-evaluated)

Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1], [0], No, outside-codomain, not-evaluated).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-changed-codomain-qualification · candidate

29. Changed Codomain Must Contain All Original Outputs

Finite types start at 0: Fin(2)={0,…,1}→Fin(3), complete images=[0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).

View answer and explanation

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

Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1], [0, 1], Yes, [], Yes).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-changed-codomain-qualification · candidate

30. Changed Codomain Must Contain All Original Outputs

Finite types start at 0: Fin(2)={0,…,1}→Fin(3), complete images=[0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).

View answer and explanation

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

Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1], [0, 1, 2], Yes, [2], No).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-changed-codomain-qualification · candidate

31. Changed Codomain Must Contain All Original Outputs

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).

View answer and explanation

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

Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 0, 0, 1], [0, 1, 2], Yes, [2], No).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-changed-codomain-qualification · candidate

32. Changed Codomain Must Contain All Original Outputs

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).

View answer and explanation

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

Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 0, 0, 1], [0, 1, 2], Yes, [2], No).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-changed-codomain-qualification · candidate

33. Changed Codomain Must Contain All Original Outputs

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).

View answer and explanation

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

Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 0, 0, 1], [0, 1, 2], Yes, [], Yes).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-changed-codomain-qualification · candidate

34. Changed Codomain Must Contain All Original Outputs

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).

View answer and explanation

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

Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 0, 0, 1], [0, 1, 2], Yes, [2], No).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-changed-codomain-qualification · candidate

35. Changed Codomain Must Contain All Original Outputs

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).

View answer and explanation

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

Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 0, 0, 1], [0, 1, 2], Yes, [2], No).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-changed-codomain-qualification · candidate

36. Changed Codomain Must Contain All Original Outputs

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).

View answer and explanation

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

Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 0, 0, 1], [0, 1, 2], Yes, [], Yes).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-changed-codomain-qualification · candidate

37. Changed Codomain Must Contain All Original Outputs

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 2, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).

View answer and explanation

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

Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 0, 0, 1], [0, 1, 2], Yes, [], Yes).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-changed-codomain-qualification · candidate

38. Changed Codomain Must Contain All Original Outputs

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 2, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).

View answer and explanation

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

Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 0, 0, 1], [0, 1, 2], Yes, [], Yes).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-changed-codomain-qualification · candidate

39. Changed Codomain Must Contain All Original Outputs

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 2, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).

View answer and explanation

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

Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2, 0, 0, 1], [0, 1, 2], Yes, [], Yes).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-changed-codomain-qualification · candidate

40. Changed Codomain Must Contain All Original Outputs

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).

View answer and explanation

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

Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1, 0, 1], [0, 1, 2], Yes, [2], No).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-changed-codomain-qualification · candidate

41. Changed Codomain Must Contain All Original Outputs

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).

View answer and explanation

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

Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 1, 0, 1], [0, 1, 2], Yes, [2], No).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-changed-codomain-qualification · candidate

42. Changed Codomain Must Contain All Original Outputs

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 0, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).

View answer and explanation

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

Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 1, 0, 1], [0, 1, 2], Yes, [], Yes).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-changed-codomain-qualification · candidate

43. Changed Codomain Must Contain All Original Outputs

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 1, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).

View answer and explanation

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

Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 1, 0, 1], [0, 1, 2], Yes, [2], No).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-changed-codomain-qualification · candidate

44. Changed Codomain Must Contain All Original Outputs

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 1, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).

View answer and explanation

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

Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 1, 0, 1], [0, 1, 2], Yes, [2], No).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-changed-codomain-qualification · candidate

45. Changed Codomain Must Contain All Original Outputs

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 1, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).

View answer and explanation

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

Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 1, 0, 1], [0, 1, 2], Yes, [], Yes).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-changed-codomain-qualification · candidate

46. Changed Codomain Must Contain All Original Outputs

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 2, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).

View answer and explanation

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

Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 1, 0, 1], [0, 1, 2], Yes, [], Yes).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-changed-codomain-qualification · candidate

47. Changed Codomain Must Contain All Original Outputs

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 2, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).

View answer and explanation

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

Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 1, 0, 1], [0, 1, 2], Yes, [], Yes).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-changed-codomain-qualification · candidate

48. Changed Codomain Must Contain All Original Outputs

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 2, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).

View answer and explanation

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

Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2, 1, 0, 1], [0, 1, 2], Yes, [], Yes).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-changed-codomain-qualification · candidate

49. Changed Codomain Must Contain All Original Outputs

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 2, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).

View answer and explanation

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

Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 2, 0, 1], [0, 1, 2], Yes, [], Yes).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-changed-codomain-qualification · candidate

50. Changed Codomain Must Contain All Original Outputs

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 2, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).

View answer and explanation

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

Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 2, 0, 1], [0, 1, 2], Yes, [], Yes).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-changed-codomain-qualification · candidate

51. Source Deletion with Declared Codomain Kept

Finite types start at 0: Fin(1)={0,…,0}→Fin(1), complete images=[0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=0 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).

View answer and explanation

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

Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0], [], [], [], [0], No).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-delete-input-missing · candidate

52. Source Deletion with Declared Codomain Kept

Finite types start at 0: Fin(3)={0,…,2}→Fin(2), complete images=[0, 1, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=1 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).

View answer and explanation

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

Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 1], [], [0, 2], [0, 1], [], Yes).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-delete-input-missing · candidate

53. Source Deletion with Declared Codomain Kept

Finite types start at 0: Fin(3)={0,…,2}→Fin(3), complete images=[0, 1, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=2 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).

View answer and explanation

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

Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 1], [2], [0, 1], [0, 1], [2], No).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-delete-input-missing · candidate

54. Source Deletion with Declared Codomain Kept

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=0 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).

View answer and explanation

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

Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 0, 0, 1], [2], [1, 2, 3, 4], [0, 0, 0, 1], [2], No).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-delete-input-missing · candidate

55. Source Deletion with Declared Codomain Kept

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=1 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).

View answer and explanation

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

Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 0, 0, 1], [2], [0, 2, 3, 4], [1, 0, 0, 1], [2], No).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-delete-input-missing · candidate

56. Source Deletion with Declared Codomain Kept

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=2 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).

View answer and explanation

([2, 0, 0, 0, 1], [], [0, 1, 3, 4], [2, 0, 0, 1], [], Yes)

Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 0, 0, 1], [], [0, 1, 3, 4], [2, 0, 0, 1], [], Yes).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-delete-input-missing · candidate

57. Source Deletion with Declared Codomain Kept

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=3 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).

View answer and explanation

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

Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 0, 0, 1], [2], [0, 1, 2, 4], [0, 1, 0, 1], [2], No).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-delete-input-missing · candidate

58. Source Deletion with Declared Codomain Kept

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=4 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).

View answer and explanation

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

Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 0, 0, 1], [2], [0, 1, 2, 3], [1, 1, 0, 0], [2], No).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-delete-input-missing · candidate

59. Source Deletion with Declared Codomain Kept

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=0 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).

View answer and explanation

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

Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 0, 0, 1], [], [1, 2, 3, 4], [1, 0, 0, 1], [2], No).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-delete-input-missing · candidate

60. Source Deletion with Declared Codomain Kept

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 2, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=1 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).

View answer and explanation

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

Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 0, 0, 1], [], [0, 2, 3, 4], [0, 0, 0, 1], [2], No).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-delete-input-missing · candidate

61. Source Deletion with Declared Codomain Kept

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 2, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=2 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).

View answer and explanation

([1, 2, 0, 0, 1], [], [0, 1, 3, 4], [1, 2, 0, 1], [], Yes)

Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 0, 0, 1], [], [0, 1, 3, 4], [1, 2, 0, 1], [], Yes).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-delete-input-missing · candidate

62. Source Deletion with Declared Codomain Kept

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 2, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=3 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).

View answer and explanation

([2, 2, 0, 0, 1], [], [0, 1, 2, 4], [2, 2, 0, 1], [], Yes)

Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2, 0, 0, 1], [], [0, 1, 2, 4], [2, 2, 0, 1], [], Yes).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-delete-input-missing · candidate

63. Source Deletion with Declared Codomain Kept

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=4 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).

View answer and explanation

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

Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. 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, 3], [0, 0, 1, 0], [2], No).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-delete-input-missing · candidate

64. Source Deletion with Declared Codomain Kept

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=0 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).

View answer and explanation

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

Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 1, 0, 1], [2], [1, 2, 3, 4], [0, 1, 0, 1], [2], No).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-delete-input-missing · candidate

65. Source Deletion with Declared Codomain Kept

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 0, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=1 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).

View answer and explanation

([2, 0, 1, 0, 1], [], [0, 2, 3, 4], [2, 1, 0, 1], [], Yes)

Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 1, 0, 1], [], [0, 2, 3, 4], [2, 1, 0, 1], [], Yes).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-delete-input-missing · candidate

66. Source Deletion with Declared Codomain Kept

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 1, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=2 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).

View answer and explanation

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

Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 1, 0, 1], [2], [0, 1, 3, 4], [0, 1, 0, 1], [2], No).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-delete-input-missing · candidate

67. Source Deletion with Declared Codomain Kept

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 1, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=3 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).

View answer and explanation

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

Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 1, 0, 1], [2], [0, 1, 2, 4], [1, 1, 1, 1], [0, 2], No).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-delete-input-missing · candidate

68. Source Deletion with Declared Codomain Kept

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 1, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=4 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).

View answer and explanation

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

Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 1, 0, 1], [], [0, 1, 2, 3], [2, 1, 1, 0], [], Yes).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-delete-input-missing · candidate

69. Source Deletion with Declared Codomain Kept

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 2, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=0 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).

View answer and explanation

([0, 2, 1, 0, 1], [], [1, 2, 3, 4], [2, 1, 0, 1], [], Yes)

Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 1, 0, 1], [], [1, 2, 3, 4], [2, 1, 0, 1], [], Yes).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-delete-input-missing · candidate

70. Source Deletion with Declared Codomain Kept

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 2, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=1 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).

View answer and explanation

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

Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 1, 0, 1], [], [0, 2, 3, 4], [1, 1, 0, 1], [2], No).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-delete-input-missing · candidate

71. Source Deletion with Declared Codomain Kept

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 2, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=2 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).

View answer and explanation

([2, 2, 1, 0, 1], [], [0, 1, 3, 4], [2, 2, 0, 1], [], Yes)

Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2, 1, 0, 1], [], [0, 1, 3, 4], [2, 2, 0, 1], [], Yes).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-delete-input-missing · candidate

72. Source Deletion with Declared Codomain Kept

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 2, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=3 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).

View answer and explanation

([0, 0, 2, 0, 1], [], [0, 1, 2, 4], [0, 0, 2, 1], [], Yes)

Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 2, 0, 1], [], [0, 1, 2, 4], [0, 0, 2, 1], [], Yes).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-delete-input-missing · candidate

73. Source Deletion with Declared Codomain Kept

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 2, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=4 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).

View answer and explanation

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

Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 2, 0, 1], [], [0, 1, 2, 3], [1, 0, 2, 0], [], Yes).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-delete-input-missing · candidate

74. Source Deletion with Declared Codomain Kept

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 0, 2, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=0 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).

View answer and explanation

([2, 0, 2, 0, 1], [], [1, 2, 3, 4], [0, 2, 0, 1], [], Yes)

Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 2, 0, 1], [], [1, 2, 3, 4], [0, 2, 0, 1], [], Yes).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-delete-input-missing · candidate

75. Source Deletion with Declared Codomain Kept

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 1, 2, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=1 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).

View answer and explanation

([0, 1, 2, 0, 1], [], [0, 2, 3, 4], [0, 2, 0, 1], [], Yes)

Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2, 0, 1], [], [0, 2, 3, 4], [0, 2, 0, 1], [], Yes).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-delete-input-missing · candidate

76. Two Target-Type Guards, Complete Fibers, and Least Preimages

Finite types start at 0: Fin(0)=∅→Fin(0), complete images=[]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, -1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).

View answer and explanation

([], [No, No], [outside-codomain, outside-codomain], [outside-codomain, outside-codomain])

Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [No, No], [outside-codomain, outside-codomain], [outside-codomain, outside-codomain]).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-two-target-preimages · candidate

77. Two Target-Type Guards, Complete Fibers, and Least Preimages

Finite types start at 0: Fin(0)=∅→Fin(2), complete images=[]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).

View answer and explanation

([], [Yes, Yes], [[], []], [does-not-exist, does-not-exist])

Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [Yes, Yes], [[], []], [does-not-exist, does-not-exist]).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-two-target-preimages · candidate

78. Two Target-Type Guards, Complete Fibers, and Least Preimages

Finite types start at 0: Fin(3)={0,…,2}→Fin(3), complete images=[0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[2, 3]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).

View answer and explanation

([0, 0, 1], [Yes, No], [[], outside-codomain], [does-not-exist, outside-codomain])

Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1], [Yes, No], [[], outside-codomain], [does-not-exist, outside-codomain]).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-two-target-preimages · candidate

79. Two Target-Type Guards, Complete Fibers, and Least Preimages

Finite types start at 0: Fin(3)={0,…,2}→Fin(2), complete images=[0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).

View answer and explanation

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

Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1], [Yes, Yes], [[0, 1], [2]], [0, 2]).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-two-target-preimages · candidate

80. Two Target-Type Guards, Complete Fibers, and Least Preimages

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 0, 0, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).

View answer and explanation

([0, 0, 0, 0, 0], [Yes, Yes], [[0, 1, 2, 3, 4], []], [0, does-not-exist])

Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 0, 0, 0], [Yes, Yes], [[0, 1, 2, 3, 4], []], [0, does-not-exist]).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-two-target-preimages · candidate

81. Two Target-Type Guards, Complete Fibers, and Least Preimages

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 0, 0, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).

View answer and explanation

([1, 0, 0, 0, 0], [Yes, Yes], [[1, 2, 3, 4], [0]], [1, 0])

Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 0, 0, 0], [Yes, Yes], [[1, 2, 3, 4], [0]], [1, 0]).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-two-target-preimages · candidate

82. Two Target-Type Guards, Complete Fibers, and Least Preimages

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 1, 0, 0, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).

View answer and explanation

([0, 1, 0, 0, 0], [Yes, Yes], [[0, 2, 3, 4], [1]], [0, 1])

Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 0, 0, 0], [Yes, Yes], [[0, 2, 3, 4], [1]], [0, 1]).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-two-target-preimages · candidate

83. Two Target-Type Guards, Complete Fibers, and Least Preimages

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 1, 0, 0, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).

View answer and explanation

([1, 1, 0, 0, 0], [Yes, Yes], [[2, 3, 4], [0, 1]], [2, 0])

Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 0, 0, 0], [Yes, Yes], [[2, 3, 4], [0, 1]], [2, 0]).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-two-target-preimages · candidate

84. Two Target-Type Guards, Complete Fibers, and Least Preimages

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 1, 0, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).

View answer and explanation

([0, 0, 1, 0, 0], [Yes, Yes], [[0, 1, 3, 4], [2]], [0, 2])

Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1, 0, 0], [Yes, Yes], [[0, 1, 3, 4], [2]], [0, 2]).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-two-target-preimages · candidate

85. Two Target-Type Guards, Complete Fibers, and Least Preimages

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 1, 0, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).

View answer and explanation

([1, 0, 1, 0, 0], [Yes, Yes], [[1, 3, 4], [0, 2]], [1, 0])

Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 1, 0, 0], [Yes, Yes], [[1, 3, 4], [0, 2]], [1, 0]).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-two-target-preimages · candidate

86. Two Target-Type Guards, Complete Fibers, and Least Preimages

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 1, 1, 0, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).

View answer and explanation

([0, 1, 1, 0, 0], [Yes, Yes], [[0, 3, 4], [1, 2]], [0, 1])

Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 1, 0, 0], [Yes, Yes], [[0, 3, 4], [1, 2]], [0, 1]).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-two-target-preimages · candidate

87. Two Target-Type Guards, Complete Fibers, and Least Preimages

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 1, 1, 0, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).

View answer and explanation

([1, 1, 1, 0, 0], [Yes, Yes], [[3, 4], [0, 1, 2]], [3, 0])

Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 1, 0, 0], [Yes, Yes], [[3, 4], [0, 1, 2]], [3, 0]).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-two-target-preimages · candidate

88. Two Target-Type Guards, Complete Fibers, and Least Preimages

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 0, 1, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).

View answer and explanation

([0, 0, 0, 1, 0], [Yes, Yes], [[0, 1, 2, 4], [3]], [0, 3])

Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 0, 1, 0], [Yes, Yes], [[0, 1, 2, 4], [3]], [0, 3]).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-two-target-preimages · candidate

89. Two Target-Type Guards, Complete Fibers, and Least Preimages

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 0, 1, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).

View answer and explanation

([1, 0, 0, 1, 0], [Yes, Yes], [[1, 2, 4], [0, 3]], [1, 0])

Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 0, 1, 0], [Yes, Yes], [[1, 2, 4], [0, 3]], [1, 0]).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-two-target-preimages · candidate

90. Two Target-Type Guards, Complete Fibers, and Least Preimages

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 1, 0, 1, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).

View answer and explanation

([0, 1, 0, 1, 0], [Yes, Yes], [[0, 2, 4], [1, 3]], [0, 1])

Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0], [Yes, Yes], [[0, 2, 4], [1, 3]], [0, 1]).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-two-target-preimages · candidate

91. Two Target-Type Guards, Complete Fibers, and Least Preimages

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 1, 0, 1, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).

View answer and explanation

([1, 1, 0, 1, 0], [Yes, Yes], [[2, 4], [0, 1, 3]], [2, 0])

Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 0, 1, 0], [Yes, Yes], [[2, 4], [0, 1, 3]], [2, 0]).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-two-target-preimages · candidate

92. Two Target-Type Guards, Complete Fibers, and Least Preimages

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 1, 1, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).

View answer and explanation

([0, 0, 1, 1, 0], [Yes, Yes], [[0, 1, 4], [2, 3]], [0, 2])

Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1, 1, 0], [Yes, Yes], [[0, 1, 4], [2, 3]], [0, 2]).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-two-target-preimages · candidate

93. Two Target-Type Guards, Complete Fibers, and Least Preimages

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 1, 1, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).

View answer and explanation

([1, 0, 1, 1, 0], [Yes, Yes], [[1, 4], [0, 2, 3]], [1, 0])

Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 1, 1, 0], [Yes, Yes], [[1, 4], [0, 2, 3]], [1, 0]).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-two-target-preimages · candidate

94. Two Target-Type Guards, Complete Fibers, and Least Preimages

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 1, 1, 1, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).

View answer and explanation

([0, 1, 1, 1, 0], [Yes, Yes], [[0, 4], [1, 2, 3]], [0, 1])

Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 1, 1, 0], [Yes, Yes], [[0, 4], [1, 2, 3]], [0, 1]).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-two-target-preimages · candidate

95. Two Target-Type Guards, Complete Fibers, and Least Preimages

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 1, 1, 1, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).

View answer and explanation

([1, 1, 1, 1, 0], [Yes, Yes], [[4], [0, 1, 2, 3]], [4, 0])

Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 1, 1, 0], [Yes, Yes], [[4], [0, 1, 2, 3]], [4, 0]).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-two-target-preimages · candidate

96. Two Target-Type Guards, Complete Fibers, and Least Preimages

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).

View answer and explanation

([0, 0, 0, 0, 1], [Yes, Yes], [[0, 1, 2, 3], [4]], [0, 4])

Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 0, 0, 1], [Yes, Yes], [[0, 1, 2, 3], [4]], [0, 4]).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-two-target-preimages · candidate

97. Two Target-Type Guards, Complete Fibers, and Least Preimages

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).

View answer and explanation

([1, 0, 0, 0, 1], [Yes, Yes], [[1, 2, 3], [0, 4]], [1, 0])

Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 0, 0, 1], [Yes, Yes], [[1, 2, 3], [0, 4]], [1, 0]).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-two-target-preimages · candidate

98. Two Target-Type Guards, Complete Fibers, and Least Preimages

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).

View answer and explanation

([0, 1, 0, 0, 1], [Yes, Yes], [[0, 2, 3], [1, 4]], [0, 1])

Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 0, 0, 1], [Yes, Yes], [[0, 2, 3], [1, 4]], [0, 1]).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-two-target-preimages · candidate

99. Two Target-Type Guards, Complete Fibers, and Least Preimages

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).

View answer and explanation

([1, 1, 0, 0, 1], [Yes, Yes], [[2, 3], [0, 1, 4]], [2, 0])

Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 0, 0, 1], [Yes, Yes], [[2, 3], [0, 1, 4]], [2, 0]).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-two-target-preimages · candidate

100. Two Target-Type Guards, Complete Fibers, and Least Preimages

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).

View answer and explanation

([0, 0, 1, 0, 1], [Yes, Yes], [[0, 1, 3], [2, 4]], [0, 2])

Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1, 0, 1], [Yes, Yes], [[0, 1, 3], [2, 4]], [0, 2]).

Reviewed step-by-step derivation

Surjectivity and codomain coverage · qf-density16-surj-two-target-preimages · candidate

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