# Injectivity Foundations

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-20261004-advance-20261005-r01-injective-foundations · 2026-10-04 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Injectivity of a Finite Total Function

The declared total function f:A→B has A=∅, B=∅, and table ∅. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.

View answer and explanationYes

Check whether each output fiber has at most one permitted input, giving Yes. Uncovered codomain elements do not defeat injectivity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0001/)

Injectivity and distinct inputs · qf-injective-table-property · candidate





## 2. Injectivity of a Finite Total Function

The declared total function f:A→B has A={-1, 0}, B={0, 1}, and table {(-1, 1), (0, 0)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.

View answer and explanationYes

Check whether each output fiber has at most one permitted input, giving Yes. Uncovered codomain elements do not defeat injectivity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0002/)

Injectivity and distinct inputs · qf-injective-table-property · candidate





## 3. Injectivity of a Finite Total Function

The declared total function f:A→B has A={-1, 0}, B={1, 3, 5}, and table {(-1, 1), (0, 3)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.

View answer and explanationYes

Check whether each output fiber has at most one permitted input, giving Yes. Uncovered codomain elements do not defeat injectivity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0003/)

Injectivity and distinct inputs · qf-injective-table-property · candidate





## 4. Injectivity of a Finite Total Function

The declared total function f:A→B has A={-1, 0}, B={0}, and table {(-1, 0), (0, 0)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.

View answer and explanationNo

Check whether each output fiber has at most one permitted input, giving No. Uncovered codomain elements do not defeat injectivity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0004/)

Injectivity and distinct inputs · qf-injective-table-property · candidate





## 5. Injectivity of a Finite Total Function

The declared total function f:A→B has A={-1, 0}, B={-3, -2, -1}, and table {(-1, -1), (0, -1)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.

View answer and explanationNo

Check whether each output fiber has at most one permitted input, giving No. Uncovered codomain elements do not defeat injectivity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0005/)

Injectivity and distinct inputs · qf-injective-table-property · candidate





## 6. Injectivity of a Finite Total Function

The declared total function f:A→B has A={-2, -1, 0}, B={0, 1, 2}, and table {(-2, 1), (-1, 2), (0, 0)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.

View answer and explanationYes

Check whether each output fiber has at most one permitted input, giving Yes. Uncovered codomain elements do not defeat injectivity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0006/)

Injectivity and distinct inputs · qf-injective-table-property · candidate





## 7. Injectivity of a Finite Total Function

The declared total function f:A→B has A={-2, -1, 0}, B={1, 3, 5, 7}, and table {(-2, 1), (-1, 3), (0, 5)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.

View answer and explanationYes

Check whether each output fiber has at most one permitted input, giving Yes. Uncovered codomain elements do not defeat injectivity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0007/)

Injectivity and distinct inputs · qf-injective-table-property · candidate





## 8. Injectivity of a Finite Total Function

The declared total function f:A→B has A={-2, -1, 0}, B={0, 1}, and table {(-2, 0), (-1, 1), (0, 0)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.

View answer and explanationNo

Check whether each output fiber has at most one permitted input, giving No. Uncovered codomain elements do not defeat injectivity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0008/)

Injectivity and distinct inputs · qf-injective-table-property · candidate





## 9. Injectivity of a Finite Total Function

The declared total function f:A→B has A={-2, -1, 0}, B={-4, -3, -2, -1}, and table {(-2, -1), (-1, -1), (0, -2)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.

View answer and explanationNo

Check whether each output fiber has at most one permitted input, giving No. Uncovered codomain elements do not defeat injectivity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0009/)

Injectivity and distinct inputs · qf-injective-table-property · candidate





## 10. Injectivity of a Finite Total Function

The declared total function f:A→B has A={-3, -2, -1, 0}, B={0, 1, 2, 3}, and table {(-3, 1), (-2, 2), (-1, 3), (0, 0)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.

View answer and explanationYes

Check whether each output fiber has at most one permitted input, giving Yes. Uncovered codomain elements do not defeat injectivity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0010/)

Injectivity and distinct inputs · qf-injective-table-property · candidate





## 11. Injectivity of a Finite Total Function

The declared total function f:A→B has A={-3, -2, -1, 0}, B={1, 3, 5, 7, 9}, and table {(-3, 1), (-2, 3), (-1, 5), (0, 7)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.

View answer and explanationYes

Check whether each output fiber has at most one permitted input, giving Yes. Uncovered codomain elements do not defeat injectivity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0011/)

Injectivity and distinct inputs · qf-injective-table-property · candidate





## 12. Injectivity of a Finite Total Function

The declared total function f:A→B has A={-3, -2, -1, 0}, B={0, 1, 2}, and table {(-3, 0), (-2, 1), (-1, 2), (0, 0)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.

View answer and explanationNo

Check whether each output fiber has at most one permitted input, giving No. Uncovered codomain elements do not defeat injectivity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0012/)

Injectivity and distinct inputs · qf-injective-table-property · candidate





## 13. Injectivity of a Finite Total Function

The declared total function f:A→B has A={-3, -2, -1, 0}, B={-5, -4, -3, -2, -1}, and table {(-3, -1), (-2, -1), (-1, -2), (0, -2)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.

View answer and explanationNo

Check whether each output fiber has at most one permitted input, giving No. Uncovered codomain elements do not defeat injectivity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0013/)

Injectivity and distinct inputs · qf-injective-table-property · candidate





## 14. Injectivity of a Finite Total Function

The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4}, and table {(-4, 1), (-3, 2), (-2, 3), (-1, 4), (0, 0)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.

View answer and explanationYes

Check whether each output fiber has at most one permitted input, giving Yes. Uncovered codomain elements do not defeat injectivity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0014/)

Injectivity and distinct inputs · qf-injective-table-property · candidate





## 15. Injectivity of a Finite Total Function

The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={1, 3, 5, 7, 9, 11}, and table {(-4, 1), (-3, 3), (-2, 5), (-1, 7), (0, 9)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.

View answer and explanationYes

Check whether each output fiber has at most one permitted input, giving Yes. Uncovered codomain elements do not defeat injectivity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0015/)

Injectivity and distinct inputs · qf-injective-table-property · candidate





## 16. Injectivity of a Finite Total Function

The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={0, 1, 2, 3}, and table {(-4, 0), (-3, 1), (-2, 2), (-1, 3), (0, 0)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.

View answer and explanationNo

Check whether each output fiber has at most one permitted input, giving No. Uncovered codomain elements do not defeat injectivity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0016/)

Injectivity and distinct inputs · qf-injective-table-property · candidate





## 17. Injectivity of a Finite Total Function

The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={-6, -5, -4, -3, -2, -1}, and table {(-4, -1), (-3, -1), (-2, -2), (-1, -2), (0, -3)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.

View answer and explanationNo

Check whether each output fiber has at most one permitted input, giving No. Uncovered codomain elements do not defeat injectivity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0017/)

Injectivity and distinct inputs · qf-injective-table-property · candidate





## 18. Injectivity of a Finite Total Function

The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4, 5}, and table {(-5, 1), (-4, 2), (-3, 3), (-2, 4), (-1, 5), (0, 0)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.

View answer and explanationYes

Check whether each output fiber has at most one permitted input, giving Yes. Uncovered codomain elements do not defeat injectivity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0018/)

Injectivity and distinct inputs · qf-injective-table-property · candidate





## 19. Injectivity of a Finite Total Function

The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={1, 3, 5, 7, 9, 11, 13}, and table {(-5, 1), (-4, 3), (-3, 5), (-2, 7), (-1, 9), (0, 11)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.

View answer and explanationYes

Check whether each output fiber has at most one permitted input, giving Yes. Uncovered codomain elements do not defeat injectivity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0019/)

Injectivity and distinct inputs · qf-injective-table-property · candidate





## 20. Injectivity of a Finite Total Function

The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4}, and table {(-5, 0), (-4, 1), (-3, 2), (-2, 3), (-1, 4), (0, 0)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.

View answer and explanationNo

Check whether each output fiber has at most one permitted input, giving No. Uncovered codomain elements do not defeat injectivity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0020/)

Injectivity and distinct inputs · qf-injective-table-property · candidate





## 21. Injectivity of a Finite Total Function

The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={-7, -6, -5, -4, -3, -2, -1}, and table {(-5, -1), (-4, -1), (-3, -2), (-2, -2), (-1, -3), (0, -3)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.

View answer and explanationNo

Check whether each output fiber has at most one permitted input, giving No. Uncovered codomain elements do not defeat injectivity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0021/)

Injectivity and distinct inputs · qf-injective-table-property · candidate





## 22. Injectivity of a Finite Total Function

The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4, 5, 6}, and table {(-6, 1), (-5, 2), (-4, 3), (-3, 4), (-2, 5), (-1, 6), (0, 0)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.

View answer and explanationYes

Check whether each output fiber has at most one permitted input, giving Yes. Uncovered codomain elements do not defeat injectivity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0022/)

Injectivity and distinct inputs · qf-injective-table-property · candidate





## 23. Injectivity of a Finite Total Function

The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={1, 3, 5, 7, 9, 11, 13, 15}, and table {(-6, 1), (-5, 3), (-4, 5), (-3, 7), (-2, 9), (-1, 11), (0, 13)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.

View answer and explanationYes

Check whether each output fiber has at most one permitted input, giving Yes. Uncovered codomain elements do not defeat injectivity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0023/)

Injectivity and distinct inputs · qf-injective-table-property · candidate





## 24. Injectivity of a Finite Total Function

The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4, 5}, and table {(-6, 0), (-5, 1), (-4, 2), (-3, 3), (-2, 4), (-1, 5), (0, 0)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.

View answer and explanationNo

Check whether each output fiber has at most one permitted input, giving No. Uncovered codomain elements do not defeat injectivity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0024/)

Injectivity and distinct inputs · qf-injective-table-property · candidate





## 25. Injectivity of a Finite Total Function

The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={-8, -7, -6, -5, -4, -3, -2, -1}, and table {(-6, -1), (-5, -1), (-4, -2), (-3, -2), (-2, -3), (-1, -3), (0, -4)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size.

View answer and explanationNo

Check whether each output fiber has at most one permitted input, giving No. Uncovered codomain elements do not defeat injectivity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0025/)

Injectivity and distinct inputs · qf-injective-table-property · candidate





## 26. Lexicographically First Collision

The declared total function f:A→B has A=∅, B=∅, and table ∅. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.

View answer and explanationnone

Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is none; repeated copies of the same pair are not distinct inputs.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0026/)

Injectivity and distinct inputs · qf-injective-first-collision · candidate





## 27. Lexicographically First Collision

The declared total function f:A→B has A={-1, 0}, B={0, 1}, and table {(-1, 1), (0, 0)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.

View answer and explanationnone

Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is none; repeated copies of the same pair are not distinct inputs.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0027/)

Injectivity and distinct inputs · qf-injective-first-collision · candidate





## 28. Lexicographically First Collision

The declared total function f:A→B has A={-1, 0}, B={1, 3, 5}, and table {(-1, 1), (0, 3)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.

View answer and explanationnone

Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is none; repeated copies of the same pair are not distinct inputs.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0028/)

Injectivity and distinct inputs · qf-injective-first-collision · candidate





## 29. Lexicographically First Collision

The declared total function f:A→B has A={-1, 0}, B={0}, and table {(-1, 0), (0, 0)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.

View answer and explanation(-1, 0)

Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is (-1, 0); repeated copies of the same pair are not distinct inputs.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0029/)

Injectivity and distinct inputs · qf-injective-first-collision · candidate





## 30. Lexicographically First Collision

The declared total function f:A→B has A={-1, 0}, B={-3, -2, -1}, and table {(-1, -1), (0, -1)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.

View answer and explanation(-1, 0)

Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is (-1, 0); repeated copies of the same pair are not distinct inputs.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0030/)

Injectivity and distinct inputs · qf-injective-first-collision · candidate





## 31. Lexicographically First Collision

The declared total function f:A→B has A={-2, -1, 0}, B={0, 1, 2}, and table {(-2, 1), (-1, 2), (0, 0)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.

View answer and explanationnone

Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is none; repeated copies of the same pair are not distinct inputs.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0031/)

Injectivity and distinct inputs · qf-injective-first-collision · candidate





## 32. Lexicographically First Collision

The declared total function f:A→B has A={-2, -1, 0}, B={1, 3, 5, 7}, and table {(-2, 1), (-1, 3), (0, 5)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.

View answer and explanationnone

Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is none; repeated copies of the same pair are not distinct inputs.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0032/)

Injectivity and distinct inputs · qf-injective-first-collision · candidate





## 33. Lexicographically First Collision

The declared total function f:A→B has A={-2, -1, 0}, B={0, 1}, and table {(-2, 0), (-1, 1), (0, 0)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.

View answer and explanation(-2, 0)

Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is (-2, 0); repeated copies of the same pair are not distinct inputs.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0033/)

Injectivity and distinct inputs · qf-injective-first-collision · candidate





## 34. Lexicographically First Collision

The declared total function f:A→B has A={-2, -1, 0}, B={-4, -3, -2, -1}, and table {(-2, -1), (-1, -1), (0, -2)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.

View answer and explanation(-2, -1)

Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is (-2, -1); repeated copies of the same pair are not distinct inputs.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0034/)

Injectivity and distinct inputs · qf-injective-first-collision · candidate





## 35. Lexicographically First Collision

The declared total function f:A→B has A={-3, -2, -1, 0}, B={0, 1, 2, 3}, and table {(-3, 1), (-2, 2), (-1, 3), (0, 0)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.

View answer and explanationnone

Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is none; repeated copies of the same pair are not distinct inputs.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0035/)

Injectivity and distinct inputs · qf-injective-first-collision · candidate





## 36. Lexicographically First Collision

The declared total function f:A→B has A={-3, -2, -1, 0}, B={1, 3, 5, 7, 9}, and table {(-3, 1), (-2, 3), (-1, 5), (0, 7)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.

View answer and explanationnone

Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is none; repeated copies of the same pair are not distinct inputs.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0036/)

Injectivity and distinct inputs · qf-injective-first-collision · candidate





## 37. Lexicographically First Collision

The declared total function f:A→B has A={-3, -2, -1, 0}, B={0, 1, 2}, and table {(-3, 0), (-2, 1), (-1, 2), (0, 0)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.

View answer and explanation(-3, 0)

Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is (-3, 0); repeated copies of the same pair are not distinct inputs.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0037/)

Injectivity and distinct inputs · qf-injective-first-collision · candidate





## 38. Lexicographically First Collision

The declared total function f:A→B has A={-3, -2, -1, 0}, B={-5, -4, -3, -2, -1}, and table {(-3, -1), (-2, -1), (-1, -2), (0, -2)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.

View answer and explanation(-3, -2)

Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is (-3, -2); repeated copies of the same pair are not distinct inputs.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0038/)

Injectivity and distinct inputs · qf-injective-first-collision · candidate





## 39. Lexicographically First Collision

The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4}, and table {(-4, 1), (-3, 2), (-2, 3), (-1, 4), (0, 0)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.

View answer and explanationnone

Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is none; repeated copies of the same pair are not distinct inputs.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0039/)

Injectivity and distinct inputs · qf-injective-first-collision · candidate





## 40. Lexicographically First Collision

The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={1, 3, 5, 7, 9, 11}, and table {(-4, 1), (-3, 3), (-2, 5), (-1, 7), (0, 9)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.

View answer and explanationnone

Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is none; repeated copies of the same pair are not distinct inputs.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0040/)

Injectivity and distinct inputs · qf-injective-first-collision · candidate





## 41. Lexicographically First Collision

The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={0, 1, 2, 3}, and table {(-4, 0), (-3, 1), (-2, 2), (-1, 3), (0, 0)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.

View answer and explanation(-4, 0)

Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is (-4, 0); repeated copies of the same pair are not distinct inputs.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0041/)

Injectivity and distinct inputs · qf-injective-first-collision · candidate





## 42. Lexicographically First Collision

The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={-6, -5, -4, -3, -2, -1}, and table {(-4, -1), (-3, -1), (-2, -2), (-1, -2), (0, -3)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.

View answer and explanation(-4, -3)

Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is (-4, -3); repeated copies of the same pair are not distinct inputs.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0042/)

Injectivity and distinct inputs · qf-injective-first-collision · candidate





## 43. Lexicographically First Collision

The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4, 5}, and table {(-5, 1), (-4, 2), (-3, 3), (-2, 4), (-1, 5), (0, 0)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.

View answer and explanationnone

Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is none; repeated copies of the same pair are not distinct inputs.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0043/)

Injectivity and distinct inputs · qf-injective-first-collision · candidate





## 44. Lexicographically First Collision

The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={1, 3, 5, 7, 9, 11, 13}, and table {(-5, 1), (-4, 3), (-3, 5), (-2, 7), (-1, 9), (0, 11)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.

View answer and explanationnone

Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is none; repeated copies of the same pair are not distinct inputs.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0044/)

Injectivity and distinct inputs · qf-injective-first-collision · candidate





## 45. Lexicographically First Collision

The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4}, and table {(-5, 0), (-4, 1), (-3, 2), (-2, 3), (-1, 4), (0, 0)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.

View answer and explanation(-5, 0)

Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is (-5, 0); repeated copies of the same pair are not distinct inputs.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0045/)

Injectivity and distinct inputs · qf-injective-first-collision · candidate





## 46. Lexicographically First Collision

The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={-7, -6, -5, -4, -3, -2, -1}, and table {(-5, -1), (-4, -1), (-3, -2), (-2, -2), (-1, -3), (0, -3)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.

View answer and explanation(-5, -4)

Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is (-5, -4); repeated copies of the same pair are not distinct inputs.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0046/)

Injectivity and distinct inputs · qf-injective-first-collision · candidate





## 47. Lexicographically First Collision

The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4, 5, 6}, and table {(-6, 1), (-5, 2), (-4, 3), (-3, 4), (-2, 5), (-1, 6), (0, 0)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.

View answer and explanationnone

Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is none; repeated copies of the same pair are not distinct inputs.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0047/)

Injectivity and distinct inputs · qf-injective-first-collision · candidate





## 48. Lexicographically First Collision

The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={1, 3, 5, 7, 9, 11, 13, 15}, and table {(-6, 1), (-5, 3), (-4, 5), (-3, 7), (-2, 9), (-1, 11), (0, 13)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.

View answer and explanationnone

Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is none; repeated copies of the same pair are not distinct inputs.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0048/)

Injectivity and distinct inputs · qf-injective-first-collision · candidate





## 49. Lexicographically First Collision

The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4, 5}, and table {(-6, 0), (-5, 1), (-4, 2), (-3, 3), (-2, 4), (-1, 5), (0, 0)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.

View answer and explanation(-6, 0)

Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is (-6, 0); repeated copies of the same pair are not distinct inputs.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0049/)

Injectivity and distinct inputs · qf-injective-first-collision · candidate





## 50. Lexicographically First Collision

The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={-8, -7, -6, -5, -4, -3, -2, -1}, and table {(-6, -1), (-5, -1), (-4, -2), (-3, -2), (-2, -3), (-1, -3), (0, -4)}. Find the first pair x<x′ with f(x)=f(x′), ordered first by x and then x′. Answer none if no collision exists.

View answer and explanation(-6, -5)

Group permitted inputs by equal output and order the distinct-input pairs numerically and lexicographically. The first collision is (-6, -5); repeated copies of the same pair are not distinct inputs.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0050/)

Injectivity and distinct inputs · qf-injective-first-collision · candidate





## 51. Cardinality of a Specified Fiber

The declared total function f:A→B has A=∅, B={0}, and table ∅. For target t=0∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.

View answer and explanation0

Enumerate inputs in A whose output is exactly 0. The fiber cardinality is 0; it is neither an output value nor a probability.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0051/)

Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate





## 52. Cardinality of a Specified Fiber

The declared total function f:A→B has A={-1, 0}, B={0, 1}, and table {(-1, 1), (0, 0)}. For target t=1∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.

View answer and explanation1

Enumerate inputs in A whose output is exactly 1. The fiber cardinality is 1; it is neither an output value nor a probability.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0052/)

Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate





## 53. Cardinality of a Specified Fiber

The declared total function f:A→B has A={-1, 0}, B={1, 3, 5}, and table {(-1, 1), (0, 3)}. For target t=5∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.

View answer and explanation0

Enumerate inputs in A whose output is exactly 5. The fiber cardinality is 0; it is neither an output value nor a probability.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0053/)

Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate





## 54. Cardinality of a Specified Fiber

The declared total function f:A→B has A={-1, 0}, B={0}, and table {(-1, 0), (0, 0)}. For target t=0∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.

View answer and explanation2

Enumerate inputs in A whose output is exactly 0. The fiber cardinality is 2; it is neither an output value nor a probability.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0054/)

Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate





## 55. Cardinality of a Specified Fiber

The declared total function f:A→B has A={-1, 0}, B={-3, -2, -1}, and table {(-1, -1), (0, -1)}. For target t=-2∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.

View answer and explanation0

Enumerate inputs in A whose output is exactly -2. The fiber cardinality is 0; it is neither an output value nor a probability.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0055/)

Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate





## 56. Cardinality of a Specified Fiber

The declared total function f:A→B has A={-2, -1, 0}, B={0, 1, 2}, and table {(-2, 1), (-1, 2), (0, 0)}. For target t=2∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.

View answer and explanation1

Enumerate inputs in A whose output is exactly 2. The fiber cardinality is 1; it is neither an output value nor a probability.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0056/)

Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate





## 57. Cardinality of a Specified Fiber

The declared total function f:A→B has A={-2, -1, 0}, B={1, 3, 5, 7}, and table {(-2, 1), (-1, 3), (0, 5)}. For target t=7∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.

View answer and explanation0

Enumerate inputs in A whose output is exactly 7. The fiber cardinality is 0; it is neither an output value nor a probability.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0057/)

Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate





## 58. Cardinality of a Specified Fiber

The declared total function f:A→B has A={-2, -1, 0}, B={0, 1}, and table {(-2, 0), (-1, 1), (0, 0)}. For target t=1∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.

View answer and explanation1

Enumerate inputs in A whose output is exactly 1. The fiber cardinality is 1; it is neither an output value nor a probability.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0058/)

Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate





## 59. Cardinality of a Specified Fiber

The declared total function f:A→B has A={-2, -1, 0}, B={-4, -3, -2, -1}, and table {(-2, -1), (-1, -1), (0, -2)}. For target t=-4∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.

View answer and explanation0

Enumerate inputs in A whose output is exactly -4. The fiber cardinality is 0; it is neither an output value nor a probability.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0059/)

Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate





## 60. Cardinality of a Specified Fiber

The declared total function f:A→B has A={-3, -2, -1, 0}, B={0, 1, 2, 3}, and table {(-3, 1), (-2, 2), (-1, 3), (0, 0)}. For target t=1∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.

View answer and explanation1

Enumerate inputs in A whose output is exactly 1. The fiber cardinality is 1; it is neither an output value nor a probability.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0060/)

Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate





## 61. Cardinality of a Specified Fiber

The declared total function f:A→B has A={-3, -2, -1, 0}, B={1, 3, 5, 7, 9}, and table {(-3, 1), (-2, 3), (-1, 5), (0, 7)}. For target t=9∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.

View answer and explanation0

Enumerate inputs in A whose output is exactly 9. The fiber cardinality is 0; it is neither an output value nor a probability.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0061/)

Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate





## 62. Cardinality of a Specified Fiber

The declared total function f:A→B has A={-3, -2, -1, 0}, B={0, 1, 2}, and table {(-3, 0), (-2, 1), (-1, 2), (0, 0)}. For target t=2∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.

View answer and explanation1

Enumerate inputs in A whose output is exactly 2. The fiber cardinality is 1; it is neither an output value nor a probability.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0062/)

Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate





## 63. Cardinality of a Specified Fiber

The declared total function f:A→B has A={-3, -2, -1, 0}, B={-5, -4, -3, -2, -1}, and table {(-3, -1), (-2, -1), (-1, -2), (0, -2)}. For target t=-3∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.

View answer and explanation0

Enumerate inputs in A whose output is exactly -3. The fiber cardinality is 0; it is neither an output value nor a probability.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0063/)

Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate





## 64. Cardinality of a Specified Fiber

The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4}, and table {(-4, 1), (-3, 2), (-2, 3), (-1, 4), (0, 0)}. For target t=3∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.

View answer and explanation1

Enumerate inputs in A whose output is exactly 3. The fiber cardinality is 1; it is neither an output value nor a probability.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0064/)

Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate





## 65. Cardinality of a Specified Fiber

The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={1, 3, 5, 7, 9, 11}, and table {(-4, 1), (-3, 3), (-2, 5), (-1, 7), (0, 9)}. For target t=11∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.

View answer and explanation0

Enumerate inputs in A whose output is exactly 11. The fiber cardinality is 0; it is neither an output value nor a probability.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0065/)

Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate





## 66. Cardinality of a Specified Fiber

The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={0, 1, 2, 3}, and table {(-4, 0), (-3, 1), (-2, 2), (-1, 3), (0, 0)}. For target t=3∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.

View answer and explanation1

Enumerate inputs in A whose output is exactly 3. The fiber cardinality is 1; it is neither an output value nor a probability.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0066/)

Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate





## 67. Cardinality of a Specified Fiber

The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={-6, -5, -4, -3, -2, -1}, and table {(-4, -1), (-3, -1), (-2, -2), (-1, -2), (0, -3)}. For target t=-2∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.

View answer and explanation2

Enumerate inputs in A whose output is exactly -2. The fiber cardinality is 2; it is neither an output value nor a probability.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0067/)

Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate





## 68. Cardinality of a Specified Fiber

The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4, 5}, and table {(-5, 1), (-4, 2), (-3, 3), (-2, 4), (-1, 5), (0, 0)}. For target t=5∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.

View answer and explanation1

Enumerate inputs in A whose output is exactly 5. The fiber cardinality is 1; it is neither an output value nor a probability.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0068/)

Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate





## 69. Cardinality of a Specified Fiber

The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={1, 3, 5, 7, 9, 11, 13}, and table {(-5, 1), (-4, 3), (-3, 5), (-2, 7), (-1, 9), (0, 11)}. For target t=13∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.

View answer and explanation0

Enumerate inputs in A whose output is exactly 13. The fiber cardinality is 0; it is neither an output value nor a probability.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0069/)

Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate





## 70. Cardinality of a Specified Fiber

The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4}, and table {(-5, 0), (-4, 1), (-3, 2), (-2, 3), (-1, 4), (0, 0)}. For target t=4∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.

View answer and explanation1

Enumerate inputs in A whose output is exactly 4. The fiber cardinality is 1; it is neither an output value nor a probability.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0070/)

Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate





## 71. Cardinality of a Specified Fiber

The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={-7, -6, -5, -4, -3, -2, -1}, and table {(-5, -1), (-4, -1), (-3, -2), (-2, -2), (-1, -3), (0, -3)}. For target t=-1∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.

View answer and explanation2

Enumerate inputs in A whose output is exactly -1. The fiber cardinality is 2; it is neither an output value nor a probability.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0071/)

Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate





## 72. Cardinality of a Specified Fiber

The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4, 5, 6}, and table {(-6, 1), (-5, 2), (-4, 3), (-3, 4), (-2, 5), (-1, 6), (0, 0)}. For target t=0∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.

View answer and explanation1

Enumerate inputs in A whose output is exactly 0. The fiber cardinality is 1; it is neither an output value nor a probability.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0072/)

Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate





## 73. Cardinality of a Specified Fiber

The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={1, 3, 5, 7, 9, 11, 13, 15}, and table {(-6, 1), (-5, 3), (-4, 5), (-3, 7), (-2, 9), (-1, 11), (0, 13)}. For target t=15∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.

View answer and explanation0

Enumerate inputs in A whose output is exactly 15. The fiber cardinality is 0; it is neither an output value nor a probability.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0073/)

Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate





## 74. Cardinality of a Specified Fiber

The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4, 5}, and table {(-6, 0), (-5, 1), (-4, 2), (-3, 3), (-2, 4), (-1, 5), (0, 0)}. For target t=5∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.

View answer and explanation1

Enumerate inputs in A whose output is exactly 5. The fiber cardinality is 1; it is neither an output value nor a probability.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0074/)

Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate





## 75. Cardinality of a Specified Fiber

The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={-8, -7, -6, -5, -4, -3, -2, -1}, and table {(-6, -1), (-5, -1), (-4, -2), (-3, -2), (-2, -3), (-1, -3), (0, -4)}. For target t=-8∈B, find the cardinality of its preimage fiber, including a zero-size fiber. The target is not an input.

View answer and explanation0

Enumerate inputs in A whose output is exactly -8. The fiber cardinality is 0; it is neither an output value nor a probability.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0075/)

Injectivity and distinct inputs · qf-injective-fiber-cardinality · candidate





## 76. Injectivity after Domain Restriction

The declared total function f:A→B has A={0, 1}, B={0}, and table {(0, 0), (1, 0)}. For C=∅⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.

View answer and explanationYes

Compare outputs of distinct inputs in C only, obtaining Yes. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0076/)

Injectivity and distinct inputs · qf-injective-restriction-property · candidate





## 77. Injectivity after Domain Restriction

The declared total function f:A→B has A={-1, 0}, B={0, 1}, and table {(-1, 1), (0, 0)}. For C={0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.

View answer and explanationYes

Compare outputs of distinct inputs in C only, obtaining Yes. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0077/)

Injectivity and distinct inputs · qf-injective-restriction-property · candidate





## 78. Injectivity after Domain Restriction

The declared total function f:A→B has A={-1, 0}, B={1, 3, 5}, and table {(-1, 1), (0, 3)}. For C={0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.

View answer and explanationYes

Compare outputs of distinct inputs in C only, obtaining Yes. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0078/)

Injectivity and distinct inputs · qf-injective-restriction-property · candidate





## 79. Injectivity after Domain Restriction

The declared total function f:A→B has A={-1, 0}, B={0}, and table {(-1, 0), (0, 0)}. For C={-1, 0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.

View answer and explanationNo

Compare outputs of distinct inputs in C only, obtaining No. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0079/)

Injectivity and distinct inputs · qf-injective-restriction-property · candidate





## 80. Injectivity after Domain Restriction

The declared total function f:A→B has A={-1, 0}, B={-3, -2, -1}, and table {(-1, -1), (0, -1)}. For C={0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.

View answer and explanationYes

Compare outputs of distinct inputs in C only, obtaining Yes. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0080/)

Injectivity and distinct inputs · qf-injective-restriction-property · candidate





## 81. Injectivity after Domain Restriction

The declared total function f:A→B has A={-2, -1, 0}, B={0, 1, 2}, and table {(-2, 1), (-1, 2), (0, 0)}. For C={-1, 0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.

View answer and explanationYes

Compare outputs of distinct inputs in C only, obtaining Yes. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0081/)

Injectivity and distinct inputs · qf-injective-restriction-property · candidate





## 82. Injectivity after Domain Restriction

The declared total function f:A→B has A={-2, -1, 0}, B={1, 3, 5, 7}, and table {(-2, 1), (-1, 3), (0, 5)}. For C={-2, -1, 0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.

View answer and explanationYes

Compare outputs of distinct inputs in C only, obtaining Yes. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0082/)

Injectivity and distinct inputs · qf-injective-restriction-property · candidate





## 83. Injectivity after Domain Restriction

The declared total function f:A→B has A={-2, -1, 0}, B={0, 1}, and table {(-2, 0), (-1, 1), (0, 0)}. For C={-1, 0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.

View answer and explanationYes

Compare outputs of distinct inputs in C only, obtaining Yes. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0083/)

Injectivity and distinct inputs · qf-injective-restriction-property · candidate





## 84. Injectivity after Domain Restriction

The declared total function f:A→B has A={-2, -1, 0}, B={-4, -3, -2, -1}, and table {(-2, -1), (-1, -1), (0, -2)}. For C={-1, 0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.

View answer and explanationYes

Compare outputs of distinct inputs in C only, obtaining Yes. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0084/)

Injectivity and distinct inputs · qf-injective-restriction-property · candidate





## 85. Injectivity after Domain Restriction

The declared total function f:A→B has A={-3, -2, -1, 0}, B={0, 1, 2, 3}, and table {(-3, 1), (-2, 2), (-1, 3), (0, 0)}. For C={-3, -2, -1, 0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.

View answer and explanationYes

Compare outputs of distinct inputs in C only, obtaining Yes. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0085/)

Injectivity and distinct inputs · qf-injective-restriction-property · candidate





## 86. Injectivity after Domain Restriction

The declared total function f:A→B has A={-3, -2, -1, 0}, B={1, 3, 5, 7, 9}, and table {(-3, 1), (-2, 3), (-1, 5), (0, 7)}. For C={-2, -1, 0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.

View answer and explanationYes

Compare outputs of distinct inputs in C only, obtaining Yes. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0086/)

Injectivity and distinct inputs · qf-injective-restriction-property · candidate





## 87. Injectivity after Domain Restriction

The declared total function f:A→B has A={-3, -2, -1, 0}, B={0, 1, 2}, and table {(-3, 0), (-2, 1), (-1, 2), (0, 0)}. For C={-2, -1, 0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.

View answer and explanationYes

Compare outputs of distinct inputs in C only, obtaining Yes. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0087/)

Injectivity and distinct inputs · qf-injective-restriction-property · candidate





## 88. Injectivity after Domain Restriction

The declared total function f:A→B has A={-3, -2, -1, 0}, B={-5, -4, -3, -2, -1}, and table {(-3, -1), (-2, -1), (-1, -2), (0, -2)}. For C={-3, -2, -1, 0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.

View answer and explanationNo

Compare outputs of distinct inputs in C only, obtaining No. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0088/)

Injectivity and distinct inputs · qf-injective-restriction-property · candidate





## 89. Injectivity after Domain Restriction

The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4}, and table {(-4, 1), (-3, 2), (-2, 3), (-1, 4), (0, 0)}. For C={-3, -2, -1, 0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.

View answer and explanationYes

Compare outputs of distinct inputs in C only, obtaining Yes. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0089/)

Injectivity and distinct inputs · qf-injective-restriction-property · candidate





## 90. Injectivity after Domain Restriction

The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={1, 3, 5, 7, 9, 11}, and table {(-4, 1), (-3, 3), (-2, 5), (-1, 7), (0, 9)}. For C={-3, -2, -1, 0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.

View answer and explanationYes

Compare outputs of distinct inputs in C only, obtaining Yes. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0090/)

Injectivity and distinct inputs · qf-injective-restriction-property · candidate





## 91. Injectivity after Domain Restriction

The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={0, 1, 2, 3}, and table {(-4, 0), (-3, 1), (-2, 2), (-1, 3), (0, 0)}. For C={-4, -3, -2, -1, 0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.

View answer and explanationNo

Compare outputs of distinct inputs in C only, obtaining No. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0091/)

Injectivity and distinct inputs · qf-injective-restriction-property · candidate





## 92. Injectivity after Domain Restriction

The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={-6, -5, -4, -3, -2, -1}, and table {(-4, -1), (-3, -1), (-2, -2), (-1, -2), (0, -3)}. For C={-3, -2, -1, 0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.

View answer and explanationNo

Compare outputs of distinct inputs in C only, obtaining No. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0092/)

Injectivity and distinct inputs · qf-injective-restriction-property · candidate





## 93. Injectivity after Domain Restriction

The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4, 5}, and table {(-5, 1), (-4, 2), (-3, 3), (-2, 4), (-1, 5), (0, 0)}. For C={-4, -3, -2, -1, 0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.

View answer and explanationYes

Compare outputs of distinct inputs in C only, obtaining Yes. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0093/)

Injectivity and distinct inputs · qf-injective-restriction-property · candidate





## 94. Injectivity after Domain Restriction

The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={1, 3, 5, 7, 9, 11, 13}, and table {(-5, 1), (-4, 3), (-3, 5), (-2, 7), (-1, 9), (0, 11)}. For C={-5, -4, -3, -2, -1, 0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.

View answer and explanationYes

Compare outputs of distinct inputs in C only, obtaining Yes. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0094/)

Injectivity and distinct inputs · qf-injective-restriction-property · candidate





## 95. Injectivity after Domain Restriction

The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4}, and table {(-5, 0), (-4, 1), (-3, 2), (-2, 3), (-1, 4), (0, 0)}. For C={-4, -3, -2, -1, 0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.

View answer and explanationYes

Compare outputs of distinct inputs in C only, obtaining Yes. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0095/)

Injectivity and distinct inputs · qf-injective-restriction-property · candidate





## 96. Injectivity after Domain Restriction

The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={-7, -6, -5, -4, -3, -2, -1}, and table {(-5, -1), (-4, -1), (-3, -2), (-2, -2), (-1, -3), (0, -3)}. For C={-4, -3, -2, -1, 0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.

View answer and explanationNo

Compare outputs of distinct inputs in C only, obtaining No. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0096/)

Injectivity and distinct inputs · qf-injective-restriction-property · candidate





## 97. Injectivity after Domain Restriction

The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4, 5, 6}, and table {(-6, 1), (-5, 2), (-4, 3), (-3, 4), (-2, 5), (-1, 6), (0, 0)}. For C={-6, -5, -4, -3, -2, -1, 0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.

View answer and explanationYes

Compare outputs of distinct inputs in C only, obtaining Yes. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0097/)

Injectivity and distinct inputs · qf-injective-restriction-property · candidate





## 98. Injectivity after Domain Restriction

The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={1, 3, 5, 7, 9, 11, 13, 15}, and table {(-6, 1), (-5, 3), (-4, 5), (-3, 7), (-2, 9), (-1, 11), (0, 13)}. For C={-5, -4, -3, -2, -1, 0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.

View answer and explanationYes

Compare outputs of distinct inputs in C only, obtaining Yes. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0098/)

Injectivity and distinct inputs · qf-injective-restriction-property · candidate





## 99. Injectivity after Domain Restriction

The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4, 5}, and table {(-6, 0), (-5, 1), (-4, 2), (-3, 3), (-2, 4), (-1, 5), (0, 0)}. For C={-5, -4, -3, -2, -1, 0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.

View answer and explanationYes

Compare outputs of distinct inputs in C only, obtaining Yes. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0099/)

Injectivity and distinct inputs · qf-injective-restriction-property · candidate





## 100. Injectivity after Domain Restriction

The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={-8, -7, -6, -5, -4, -3, -2, -1}, and table {(-6, -1), (-5, -1), (-4, -2), (-3, -2), (-2, -3), (-1, -3), (0, -4)}. For C={-6, -5, -4, -3, -2, -1, 0}⊆A, is the restriction f|C:C→B injective? Consider only the restricted function.

View answer and explanationNo

Compare outputs of distinct inputs in C only, obtaining No. Collisions outside C do not count, and empty C satisfies the condition. This does not establish injectivity of the original function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r01-injective-0100/)

Injectivity and distinct inputs · qf-injective-restriction-property · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261004-advance-20261005-r01-injective-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261004-advance-20261005-r01-injective-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261004-advance-20261005-r01-injective-foundations/validation-report.json)