# Declared Codomain Fibers, Changes, and Target Guards

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

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

batch-20261012-advance-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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0001/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0002/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0003/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0004/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0005/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0006/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0007/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0008/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0009/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0010/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0011/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0012/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0013/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0014/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0015/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0016/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0017/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0018/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0019/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0020/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0021/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0022/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0023/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0024/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0025/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0026/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0027/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0028/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0029/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0030/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0031/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0032/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0033/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0034/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0035/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0036/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0037/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0038/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0039/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0040/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0041/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0042/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0043/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0044/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0045/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0046/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0047/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0048/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0049/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0050/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0051/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0052/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0053/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0054/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0055/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0056/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0057/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0058/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0059/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0060/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0061/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0062/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0063/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0064/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0065/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0066/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0067/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0068/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0069/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0070/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0071/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0072/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0073/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0074/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0075/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0076/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0077/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0078/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0079/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0080/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0081/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0082/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0083/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0084/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0085/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0086/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0087/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0088/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0089/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0090/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0091/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0092/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0093/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0094/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0095/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0096/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0097/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0098/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0099/)

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](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r07-surjective-0100/)

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



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261012-advance-20261016-r07-surjective-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261012-advance-20261016-r07-surjective-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261012-advance-20261016-r07-surjective-foundations/validation-report.json)