# Bijectivity 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-r03-bijective-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. Bijectivity of a Finite Total Function

The declared total function f:A→B has A=∅, B=∅, and table ∅. Is f both injective and surjective? Decide bijectivity from the actual table.

View answer and explanationYes

Both conditions must hold, equivalently every codomain target must have exactly one permitted preimage. The answer is Yes.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0001/)

Bijectivity and unique preimages · qf-bijective-table-property · candidate





## 2. Bijectivity 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 both injective and surjective? Decide bijectivity from the actual table.

View answer and explanationYes

Both conditions must hold, equivalently every codomain target must have exactly one permitted preimage. The answer is Yes.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0002/)

Bijectivity and unique preimages · qf-bijective-table-property · candidate





## 3. Bijectivity 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 both injective and surjective? Decide bijectivity from the actual table.

View answer and explanationNo

Both conditions must hold, equivalently every codomain target must have exactly one permitted preimage. The answer is No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0003/)

Bijectivity and unique preimages · qf-bijective-table-property · candidate





## 4. Bijectivity 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 both injective and surjective? Decide bijectivity from the actual table.

View answer and explanationNo

Both conditions must hold, equivalently every codomain target must have exactly one permitted preimage. The answer is No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0004/)

Bijectivity and unique preimages · qf-bijective-table-property · candidate





## 5. Bijectivity 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 both injective and surjective? Decide bijectivity from the actual table.

View answer and explanationNo

Both conditions must hold, equivalently every codomain target must have exactly one permitted preimage. The answer is No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0005/)

Bijectivity and unique preimages · qf-bijective-table-property · candidate





## 6. Bijectivity 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 both injective and surjective? Decide bijectivity from the actual table.

View answer and explanationYes

Both conditions must hold, equivalently every codomain target must have exactly one permitted preimage. The answer is Yes.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0006/)

Bijectivity and unique preimages · qf-bijective-table-property · candidate





## 7. Bijectivity 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 both injective and surjective? Decide bijectivity from the actual table.

View answer and explanationNo

Both conditions must hold, equivalently every codomain target must have exactly one permitted preimage. The answer is No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0007/)

Bijectivity and unique preimages · qf-bijective-table-property · candidate





## 8. Bijectivity 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 both injective and surjective? Decide bijectivity from the actual table.

View answer and explanationNo

Both conditions must hold, equivalently every codomain target must have exactly one permitted preimage. The answer is No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0008/)

Bijectivity and unique preimages · qf-bijective-table-property · candidate





## 9. Bijectivity 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 both injective and surjective? Decide bijectivity from the actual table.

View answer and explanationNo

Both conditions must hold, equivalently every codomain target must have exactly one permitted preimage. The answer is No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0009/)

Bijectivity and unique preimages · qf-bijective-table-property · candidate





## 10. Bijectivity 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 both injective and surjective? Decide bijectivity from the actual table.

View answer and explanationYes

Both conditions must hold, equivalently every codomain target must have exactly one permitted preimage. The answer is Yes.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0010/)

Bijectivity and unique preimages · qf-bijective-table-property · candidate





## 11. Bijectivity 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 both injective and surjective? Decide bijectivity from the actual table.

View answer and explanationNo

Both conditions must hold, equivalently every codomain target must have exactly one permitted preimage. The answer is No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0011/)

Bijectivity and unique preimages · qf-bijective-table-property · candidate





## 12. Bijectivity 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 both injective and surjective? Decide bijectivity from the actual table.

View answer and explanationNo

Both conditions must hold, equivalently every codomain target must have exactly one permitted preimage. The answer is No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0012/)

Bijectivity and unique preimages · qf-bijective-table-property · candidate





## 13. Bijectivity 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 both injective and surjective? Decide bijectivity from the actual table.

View answer and explanationNo

Both conditions must hold, equivalently every codomain target must have exactly one permitted preimage. The answer is No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0013/)

Bijectivity and unique preimages · qf-bijective-table-property · candidate





## 14. Bijectivity 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 both injective and surjective? Decide bijectivity from the actual table.

View answer and explanationYes

Both conditions must hold, equivalently every codomain target must have exactly one permitted preimage. The answer is Yes.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0014/)

Bijectivity and unique preimages · qf-bijective-table-property · candidate





## 15. Bijectivity 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 both injective and surjective? Decide bijectivity from the actual table.

View answer and explanationNo

Both conditions must hold, equivalently every codomain target must have exactly one permitted preimage. The answer is No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0015/)

Bijectivity and unique preimages · qf-bijective-table-property · candidate





## 16. Bijectivity 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 both injective and surjective? Decide bijectivity from the actual table.

View answer and explanationNo

Both conditions must hold, equivalently every codomain target must have exactly one permitted preimage. The answer is No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0016/)

Bijectivity and unique preimages · qf-bijective-table-property · candidate





## 17. Bijectivity 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 both injective and surjective? Decide bijectivity from the actual table.

View answer and explanationNo

Both conditions must hold, equivalently every codomain target must have exactly one permitted preimage. The answer is No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0017/)

Bijectivity and unique preimages · qf-bijective-table-property · candidate





## 18. Bijectivity 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 both injective and surjective? Decide bijectivity from the actual table.

View answer and explanationYes

Both conditions must hold, equivalently every codomain target must have exactly one permitted preimage. The answer is Yes.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0018/)

Bijectivity and unique preimages · qf-bijective-table-property · candidate





## 19. Bijectivity 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 both injective and surjective? Decide bijectivity from the actual table.

View answer and explanationNo

Both conditions must hold, equivalently every codomain target must have exactly one permitted preimage. The answer is No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0019/)

Bijectivity and unique preimages · qf-bijective-table-property · candidate





## 20. Bijectivity 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 both injective and surjective? Decide bijectivity from the actual table.

View answer and explanationNo

Both conditions must hold, equivalently every codomain target must have exactly one permitted preimage. The answer is No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0020/)

Bijectivity and unique preimages · qf-bijective-table-property · candidate





## 21. Bijectivity 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 both injective and surjective? Decide bijectivity from the actual table.

View answer and explanationNo

Both conditions must hold, equivalently every codomain target must have exactly one permitted preimage. The answer is No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0021/)

Bijectivity and unique preimages · qf-bijective-table-property · candidate





## 22. Bijectivity 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 both injective and surjective? Decide bijectivity from the actual table.

View answer and explanationYes

Both conditions must hold, equivalently every codomain target must have exactly one permitted preimage. The answer is Yes.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0022/)

Bijectivity and unique preimages · qf-bijective-table-property · candidate





## 23. Bijectivity 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 both injective and surjective? Decide bijectivity from the actual table.

View answer and explanationNo

Both conditions must hold, equivalently every codomain target must have exactly one permitted preimage. The answer is No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0023/)

Bijectivity and unique preimages · qf-bijective-table-property · candidate





## 24. Bijectivity 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 both injective and surjective? Decide bijectivity from the actual table.

View answer and explanationNo

Both conditions must hold, equivalently every codomain target must have exactly one permitted preimage. The answer is No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0024/)

Bijectivity and unique preimages · qf-bijective-table-property · candidate





## 25. Bijectivity 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 both injective and surjective? Decide bijectivity from the actual table.

View answer and explanationNo

Both conditions must hold, equivalently every codomain target must have exactly one permitted preimage. The answer is No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0025/)

Bijectivity and unique preimages · qf-bijective-table-property · candidate





## 26. Four-Way Function Classification

The declared total function f:A→B has A=∅, B=∅, and table ∅. Classify f as bijective, injective only, surjective only, or neither injective nor surjective.

View answer and explanationbijective

Check distinct-input collisions and full codomain coverage separately; their two truth values give bijective. Set sizes alone do not classify this supplied function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0026/)

Bijectivity and unique preimages · qf-bijective-classification · candidate





## 27. Four-Way Function Classification

The declared total function f:A→B has A={-1, 0}, B={0, 1}, and table {(-1, 1), (0, 0)}. Classify f as bijective, injective only, surjective only, or neither injective nor surjective.

View answer and explanationbijective

Check distinct-input collisions and full codomain coverage separately; their two truth values give bijective. Set sizes alone do not classify this supplied function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0027/)

Bijectivity and unique preimages · qf-bijective-classification · candidate





## 28. Four-Way Function Classification

The declared total function f:A→B has A={-1, 0}, B={1, 3, 5}, and table {(-1, 1), (0, 3)}. Classify f as bijective, injective only, surjective only, or neither injective nor surjective.

View answer and explanationinjective only

Check distinct-input collisions and full codomain coverage separately; their two truth values give injective only. Set sizes alone do not classify this supplied function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0028/)

Bijectivity and unique preimages · qf-bijective-classification · candidate





## 29. Four-Way Function Classification

The declared total function f:A→B has A={-1, 0}, B={0}, and table {(-1, 0), (0, 0)}. Classify f as bijective, injective only, surjective only, or neither injective nor surjective.

View answer and explanationsurjective only

Check distinct-input collisions and full codomain coverage separately; their two truth values give surjective only. Set sizes alone do not classify this supplied function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0029/)

Bijectivity and unique preimages · qf-bijective-classification · candidate





## 30. Four-Way Function Classification

The declared total function f:A→B has A={-1, 0}, B={-3, -2, -1}, and table {(-1, -1), (0, -1)}. Classify f as bijective, injective only, surjective only, or neither injective nor surjective.

View answer and explanationneither injective nor surjective

Check distinct-input collisions and full codomain coverage separately; their two truth values give neither injective nor surjective. Set sizes alone do not classify this supplied function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0030/)

Bijectivity and unique preimages · qf-bijective-classification · candidate





## 31. Four-Way Function Classification

The declared total function f:A→B has A={-2, -1, 0}, B={0, 1, 2}, and table {(-2, 1), (-1, 2), (0, 0)}. Classify f as bijective, injective only, surjective only, or neither injective nor surjective.

View answer and explanationbijective

Check distinct-input collisions and full codomain coverage separately; their two truth values give bijective. Set sizes alone do not classify this supplied function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0031/)

Bijectivity and unique preimages · qf-bijective-classification · candidate





## 32. Four-Way Function Classification

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)}. Classify f as bijective, injective only, surjective only, or neither injective nor surjective.

View answer and explanationinjective only

Check distinct-input collisions and full codomain coverage separately; their two truth values give injective only. Set sizes alone do not classify this supplied function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0032/)

Bijectivity and unique preimages · qf-bijective-classification · candidate





## 33. Four-Way Function Classification

The declared total function f:A→B has A={-2, -1, 0}, B={0, 1}, and table {(-2, 0), (-1, 1), (0, 0)}. Classify f as bijective, injective only, surjective only, or neither injective nor surjective.

View answer and explanationsurjective only

Check distinct-input collisions and full codomain coverage separately; their two truth values give surjective only. Set sizes alone do not classify this supplied function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0033/)

Bijectivity and unique preimages · qf-bijective-classification · candidate





## 34. Four-Way Function Classification

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)}. Classify f as bijective, injective only, surjective only, or neither injective nor surjective.

View answer and explanationneither injective nor surjective

Check distinct-input collisions and full codomain coverage separately; their two truth values give neither injective nor surjective. Set sizes alone do not classify this supplied function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0034/)

Bijectivity and unique preimages · qf-bijective-classification · candidate





## 35. Four-Way Function Classification

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)}. Classify f as bijective, injective only, surjective only, or neither injective nor surjective.

View answer and explanationbijective

Check distinct-input collisions and full codomain coverage separately; their two truth values give bijective. Set sizes alone do not classify this supplied function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0035/)

Bijectivity and unique preimages · qf-bijective-classification · candidate





## 36. Four-Way Function Classification

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)}. Classify f as bijective, injective only, surjective only, or neither injective nor surjective.

View answer and explanationinjective only

Check distinct-input collisions and full codomain coverage separately; their two truth values give injective only. Set sizes alone do not classify this supplied function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0036/)

Bijectivity and unique preimages · qf-bijective-classification · candidate





## 37. Four-Way Function Classification

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)}. Classify f as bijective, injective only, surjective only, or neither injective nor surjective.

View answer and explanationsurjective only

Check distinct-input collisions and full codomain coverage separately; their two truth values give surjective only. Set sizes alone do not classify this supplied function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0037/)

Bijectivity and unique preimages · qf-bijective-classification · candidate





## 38. Four-Way Function Classification

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)}. Classify f as bijective, injective only, surjective only, or neither injective nor surjective.

View answer and explanationneither injective nor surjective

Check distinct-input collisions and full codomain coverage separately; their two truth values give neither injective nor surjective. Set sizes alone do not classify this supplied function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0038/)

Bijectivity and unique preimages · qf-bijective-classification · candidate





## 39. Four-Way Function Classification

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)}. Classify f as bijective, injective only, surjective only, or neither injective nor surjective.

View answer and explanationbijective

Check distinct-input collisions and full codomain coverage separately; their two truth values give bijective. Set sizes alone do not classify this supplied function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0039/)

Bijectivity and unique preimages · qf-bijective-classification · candidate





## 40. Four-Way Function Classification

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)}. Classify f as bijective, injective only, surjective only, or neither injective nor surjective.

View answer and explanationinjective only

Check distinct-input collisions and full codomain coverage separately; their two truth values give injective only. Set sizes alone do not classify this supplied function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0040/)

Bijectivity and unique preimages · qf-bijective-classification · candidate





## 41. Four-Way Function Classification

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)}. Classify f as bijective, injective only, surjective only, or neither injective nor surjective.

View answer and explanationsurjective only

Check distinct-input collisions and full codomain coverage separately; their two truth values give surjective only. Set sizes alone do not classify this supplied function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0041/)

Bijectivity and unique preimages · qf-bijective-classification · candidate





## 42. Four-Way Function Classification

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)}. Classify f as bijective, injective only, surjective only, or neither injective nor surjective.

View answer and explanationneither injective nor surjective

Check distinct-input collisions and full codomain coverage separately; their two truth values give neither injective nor surjective. Set sizes alone do not classify this supplied function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0042/)

Bijectivity and unique preimages · qf-bijective-classification · candidate





## 43. Four-Way Function Classification

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)}. Classify f as bijective, injective only, surjective only, or neither injective nor surjective.

View answer and explanationbijective

Check distinct-input collisions and full codomain coverage separately; their two truth values give bijective. Set sizes alone do not classify this supplied function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0043/)

Bijectivity and unique preimages · qf-bijective-classification · candidate





## 44. Four-Way Function Classification

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)}. Classify f as bijective, injective only, surjective only, or neither injective nor surjective.

View answer and explanationinjective only

Check distinct-input collisions and full codomain coverage separately; their two truth values give injective only. Set sizes alone do not classify this supplied function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0044/)

Bijectivity and unique preimages · qf-bijective-classification · candidate





## 45. Four-Way Function Classification

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)}. Classify f as bijective, injective only, surjective only, or neither injective nor surjective.

View answer and explanationsurjective only

Check distinct-input collisions and full codomain coverage separately; their two truth values give surjective only. Set sizes alone do not classify this supplied function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0045/)

Bijectivity and unique preimages · qf-bijective-classification · candidate





## 46. Four-Way Function Classification

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)}. Classify f as bijective, injective only, surjective only, or neither injective nor surjective.

View answer and explanationneither injective nor surjective

Check distinct-input collisions and full codomain coverage separately; their two truth values give neither injective nor surjective. Set sizes alone do not classify this supplied function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0046/)

Bijectivity and unique preimages · qf-bijective-classification · candidate





## 47. Four-Way Function Classification

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)}. Classify f as bijective, injective only, surjective only, or neither injective nor surjective.

View answer and explanationbijective

Check distinct-input collisions and full codomain coverage separately; their two truth values give bijective. Set sizes alone do not classify this supplied function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0047/)

Bijectivity and unique preimages · qf-bijective-classification · candidate





## 48. Four-Way Function Classification

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)}. Classify f as bijective, injective only, surjective only, or neither injective nor surjective.

View answer and explanationinjective only

Check distinct-input collisions and full codomain coverage separately; their two truth values give injective only. Set sizes alone do not classify this supplied function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0048/)

Bijectivity and unique preimages · qf-bijective-classification · candidate





## 49. Four-Way Function Classification

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)}. Classify f as bijective, injective only, surjective only, or neither injective nor surjective.

View answer and explanationsurjective only

Check distinct-input collisions and full codomain coverage separately; their two truth values give surjective only. Set sizes alone do not classify this supplied function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0049/)

Bijectivity and unique preimages · qf-bijective-classification · candidate





## 50. Four-Way Function Classification

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)}. Classify f as bijective, injective only, surjective only, or neither injective nor surjective.

View answer and explanationneither injective nor surjective

Check distinct-input collisions and full codomain coverage separately; their two truth values give neither injective nor surjective. Set sizes alone do not classify this supplied function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0050/)

Bijectivity and unique preimages · qf-bijective-classification · candidate





## 51. Complete Codomain Fiber Profile

The declared total function f:A→B has A=∅, B=∅, and table ∅. In increasing target order, list every (b,|f⁻¹({b})|), including zero for targets not attained.

View answer and explanation∅

Count the permitted preimages of every element of B to obtain ∅. The profile includes zero-size fibers, not only attained targets.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0051/)

Bijectivity and unique preimages · qf-bijective-fiber-profile · candidate





## 52. Complete Codomain Fiber Profile

The declared total function f:A→B has A={-1, 0}, B={0, 1}, and table {(-1, 1), (0, 0)}. In increasing target order, list every (b,|f⁻¹({b})|), including zero for targets not attained.

View answer and explanation{(0, 1), (1, 1)}

Count the permitted preimages of every element of B to obtain {(0, 1), (1, 1)}. The profile includes zero-size fibers, not only attained targets.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0052/)

Bijectivity and unique preimages · qf-bijective-fiber-profile · candidate





## 53. Complete Codomain Fiber Profile

The declared total function f:A→B has A={-1, 0}, B={1, 3, 5}, and table {(-1, 1), (0, 3)}. In increasing target order, list every (b,|f⁻¹({b})|), including zero for targets not attained.

View answer and explanation{(1, 1), (3, 1), (5, 0)}

Count the permitted preimages of every element of B to obtain {(1, 1), (3, 1), (5, 0)}. The profile includes zero-size fibers, not only attained targets.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0053/)

Bijectivity and unique preimages · qf-bijective-fiber-profile · candidate





## 54. Complete Codomain Fiber Profile

The declared total function f:A→B has A={-1, 0}, B={0}, and table {(-1, 0), (0, 0)}. In increasing target order, list every (b,|f⁻¹({b})|), including zero for targets not attained.

View answer and explanation{(0, 2)}

Count the permitted preimages of every element of B to obtain {(0, 2)}. The profile includes zero-size fibers, not only attained targets.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0054/)

Bijectivity and unique preimages · qf-bijective-fiber-profile · candidate





## 55. Complete Codomain Fiber Profile

The declared total function f:A→B has A={-1, 0}, B={-3, -2, -1}, and table {(-1, -1), (0, -1)}. In increasing target order, list every (b,|f⁻¹({b})|), including zero for targets not attained.

View answer and explanation{(-3, 0), (-2, 0), (-1, 2)}

Count the permitted preimages of every element of B to obtain {(-3, 0), (-2, 0), (-1, 2)}. The profile includes zero-size fibers, not only attained targets.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0055/)

Bijectivity and unique preimages · qf-bijective-fiber-profile · candidate





## 56. Complete Codomain Fiber Profile

The declared total function f:A→B has A={-2, -1, 0}, B={0, 1, 2}, and table {(-2, 1), (-1, 2), (0, 0)}. In increasing target order, list every (b,|f⁻¹({b})|), including zero for targets not attained.

View answer and explanation{(0, 1), (1, 1), (2, 1)}

Count the permitted preimages of every element of B to obtain {(0, 1), (1, 1), (2, 1)}. The profile includes zero-size fibers, not only attained targets.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0056/)

Bijectivity and unique preimages · qf-bijective-fiber-profile · candidate





## 57. Complete Codomain Fiber Profile

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)}. In increasing target order, list every (b,|f⁻¹({b})|), including zero for targets not attained.

View answer and explanation{(1, 1), (3, 1), (5, 1), (7, 0)}

Count the permitted preimages of every element of B to obtain {(1, 1), (3, 1), (5, 1), (7, 0)}. The profile includes zero-size fibers, not only attained targets.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0057/)

Bijectivity and unique preimages · qf-bijective-fiber-profile · candidate





## 58. Complete Codomain Fiber Profile

The declared total function f:A→B has A={-2, -1, 0}, B={0, 1}, and table {(-2, 0), (-1, 1), (0, 0)}. In increasing target order, list every (b,|f⁻¹({b})|), including zero for targets not attained.

View answer and explanation{(0, 2), (1, 1)}

Count the permitted preimages of every element of B to obtain {(0, 2), (1, 1)}. The profile includes zero-size fibers, not only attained targets.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0058/)

Bijectivity and unique preimages · qf-bijective-fiber-profile · candidate





## 59. Complete Codomain Fiber Profile

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)}. In increasing target order, list every (b,|f⁻¹({b})|), including zero for targets not attained.

View answer and explanation{(-4, 0), (-3, 0), (-2, 1), (-1, 2)}

Count the permitted preimages of every element of B to obtain {(-4, 0), (-3, 0), (-2, 1), (-1, 2)}. The profile includes zero-size fibers, not only attained targets.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0059/)

Bijectivity and unique preimages · qf-bijective-fiber-profile · candidate





## 60. Complete Codomain Fiber Profile

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)}. In increasing target order, list every (b,|f⁻¹({b})|), including zero for targets not attained.

View answer and explanation{(0, 1), (1, 1), (2, 1), (3, 1)}

Count the permitted preimages of every element of B to obtain {(0, 1), (1, 1), (2, 1), (3, 1)}. The profile includes zero-size fibers, not only attained targets.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0060/)

Bijectivity and unique preimages · qf-bijective-fiber-profile · candidate





## 61. Complete Codomain Fiber Profile

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)}. In increasing target order, list every (b,|f⁻¹({b})|), including zero for targets not attained.

View answer and explanation{(1, 1), (3, 1), (5, 1), (7, 1), (9, 0)}

Count the permitted preimages of every element of B to obtain {(1, 1), (3, 1), (5, 1), (7, 1), (9, 0)}. The profile includes zero-size fibers, not only attained targets.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0061/)

Bijectivity and unique preimages · qf-bijective-fiber-profile · candidate





## 62. Complete Codomain Fiber Profile

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)}. In increasing target order, list every (b,|f⁻¹({b})|), including zero for targets not attained.

View answer and explanation{(0, 2), (1, 1), (2, 1)}

Count the permitted preimages of every element of B to obtain {(0, 2), (1, 1), (2, 1)}. The profile includes zero-size fibers, not only attained targets.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0062/)

Bijectivity and unique preimages · qf-bijective-fiber-profile · candidate





## 63. Complete Codomain Fiber Profile

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)}. In increasing target order, list every (b,|f⁻¹({b})|), including zero for targets not attained.

View answer and explanation{(-5, 0), (-4, 0), (-3, 0), (-2, 2), (-1, 2)}

Count the permitted preimages of every element of B to obtain {(-5, 0), (-4, 0), (-3, 0), (-2, 2), (-1, 2)}. The profile includes zero-size fibers, not only attained targets.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0063/)

Bijectivity and unique preimages · qf-bijective-fiber-profile · candidate





## 64. Complete Codomain Fiber Profile

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)}. In increasing target order, list every (b,|f⁻¹({b})|), including zero for targets not attained.

View answer and explanation{(0, 1), (1, 1), (2, 1), (3, 1), (4, 1)}

Count the permitted preimages of every element of B to obtain {(0, 1), (1, 1), (2, 1), (3, 1), (4, 1)}. The profile includes zero-size fibers, not only attained targets.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0064/)

Bijectivity and unique preimages · qf-bijective-fiber-profile · candidate





## 65. Complete Codomain Fiber Profile

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)}. In increasing target order, list every (b,|f⁻¹({b})|), including zero for targets not attained.

View answer and explanation{(1, 1), (3, 1), (5, 1), (7, 1), (9, 1), (11, 0)}

Count the permitted preimages of every element of B to obtain {(1, 1), (3, 1), (5, 1), (7, 1), (9, 1), (11, 0)}. The profile includes zero-size fibers, not only attained targets.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0065/)

Bijectivity and unique preimages · qf-bijective-fiber-profile · candidate





## 66. Complete Codomain Fiber Profile

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)}. In increasing target order, list every (b,|f⁻¹({b})|), including zero for targets not attained.

View answer and explanation{(0, 2), (1, 1), (2, 1), (3, 1)}

Count the permitted preimages of every element of B to obtain {(0, 2), (1, 1), (2, 1), (3, 1)}. The profile includes zero-size fibers, not only attained targets.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0066/)

Bijectivity and unique preimages · qf-bijective-fiber-profile · candidate





## 67. Complete Codomain Fiber Profile

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)}. In increasing target order, list every (b,|f⁻¹({b})|), including zero for targets not attained.

View answer and explanation{(-6, 0), (-5, 0), (-4, 0), (-3, 1), (-2, 2), (-1, 2)}

Count the permitted preimages of every element of B to obtain {(-6, 0), (-5, 0), (-4, 0), (-3, 1), (-2, 2), (-1, 2)}. The profile includes zero-size fibers, not only attained targets.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0067/)

Bijectivity and unique preimages · qf-bijective-fiber-profile · candidate





## 68. Complete Codomain Fiber Profile

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)}. In increasing target order, list every (b,|f⁻¹({b})|), including zero for targets not attained.

View answer and explanation{(0, 1), (1, 1), (2, 1), (3, 1), (4, 1), (5, 1)}

Count the permitted preimages of every element of B to obtain {(0, 1), (1, 1), (2, 1), (3, 1), (4, 1), (5, 1)}. The profile includes zero-size fibers, not only attained targets.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0068/)

Bijectivity and unique preimages · qf-bijective-fiber-profile · candidate





## 69. Complete Codomain Fiber Profile

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)}. In increasing target order, list every (b,|f⁻¹({b})|), including zero for targets not attained.

View answer and explanation{(1, 1), (3, 1), (5, 1), (7, 1), (9, 1), (11, 1), (13, 0)}

Count the permitted preimages of every element of B to obtain {(1, 1), (3, 1), (5, 1), (7, 1), (9, 1), (11, 1), (13, 0)}. The profile includes zero-size fibers, not only attained targets.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0069/)

Bijectivity and unique preimages · qf-bijective-fiber-profile · candidate





## 70. Complete Codomain Fiber Profile

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)}. In increasing target order, list every (b,|f⁻¹({b})|), including zero for targets not attained.

View answer and explanation{(0, 2), (1, 1), (2, 1), (3, 1), (4, 1)}

Count the permitted preimages of every element of B to obtain {(0, 2), (1, 1), (2, 1), (3, 1), (4, 1)}. The profile includes zero-size fibers, not only attained targets.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0070/)

Bijectivity and unique preimages · qf-bijective-fiber-profile · candidate





## 71. Complete Codomain Fiber Profile

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)}. In increasing target order, list every (b,|f⁻¹({b})|), including zero for targets not attained.

View answer and explanation{(-7, 0), (-6, 0), (-5, 0), (-4, 0), (-3, 2), (-2, 2), (-1, 2)}

Count the permitted preimages of every element of B to obtain {(-7, 0), (-6, 0), (-5, 0), (-4, 0), (-3, 2), (-2, 2), (-1, 2)}. The profile includes zero-size fibers, not only attained targets.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0071/)

Bijectivity and unique preimages · qf-bijective-fiber-profile · candidate





## 72. Complete Codomain Fiber Profile

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)}. In increasing target order, list every (b,|f⁻¹({b})|), including zero for targets not attained.

View answer and explanation{(0, 1), (1, 1), (2, 1), (3, 1), (4, 1), (5, 1), (6, 1)}

Count the permitted preimages of every element of B to obtain {(0, 1), (1, 1), (2, 1), (3, 1), (4, 1), (5, 1), (6, 1)}. The profile includes zero-size fibers, not only attained targets.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0072/)

Bijectivity and unique preimages · qf-bijective-fiber-profile · candidate





## 73. Complete Codomain Fiber Profile

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)}. In increasing target order, list every (b,|f⁻¹({b})|), including zero for targets not attained.

View answer and explanation{(1, 1), (3, 1), (5, 1), (7, 1), (9, 1), (11, 1), (13, 1), (15, 0)}

Count the permitted preimages of every element of B to obtain {(1, 1), (3, 1), (5, 1), (7, 1), (9, 1), (11, 1), (13, 1), (15, 0)}. The profile includes zero-size fibers, not only attained targets.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0073/)

Bijectivity and unique preimages · qf-bijective-fiber-profile · candidate





## 74. Complete Codomain Fiber Profile

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)}. In increasing target order, list every (b,|f⁻¹({b})|), including zero for targets not attained.

View answer and explanation{(0, 2), (1, 1), (2, 1), (3, 1), (4, 1), (5, 1)}

Count the permitted preimages of every element of B to obtain {(0, 2), (1, 1), (2, 1), (3, 1), (4, 1), (5, 1)}. The profile includes zero-size fibers, not only attained targets.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0074/)

Bijectivity and unique preimages · qf-bijective-fiber-profile · candidate





## 75. Complete Codomain Fiber Profile

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)}. In increasing target order, list every (b,|f⁻¹({b})|), including zero for targets not attained.

View answer and explanation{(-8, 0), (-7, 0), (-6, 0), (-5, 0), (-4, 1), (-3, 2), (-2, 2), (-1, 2)}

Count the permitted preimages of every element of B to obtain {(-8, 0), (-7, 0), (-6, 0), (-5, 0), (-4, 1), (-3, 2), (-2, 2), (-1, 2)}. The profile includes zero-size fibers, not only attained targets.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0075/)

Bijectivity and unique preimages · qf-bijective-fiber-profile · candidate





## 76. Existence of a Bijection from Finite Cardinalities

The explicit finite cardinalities are |A|=0 and |B|=0, with no particular function table supplied. Does some bijection A→B exist?

View answer and explanationYes

Finite sets admit a bijection exactly when their cardinalities agree, giving Yes. This asks about existence rather than testing a given function; two empty sets admit the unique empty bijection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0076/)

Bijectivity and unique preimages · qf-bijective-cardinality-possibility · candidate





## 77. Existence of a Bijection from Finite Cardinalities

The explicit finite cardinalities are |A|=0 and |B|=1, with no particular function table supplied. Does some bijection A→B exist?

View answer and explanationNo

Finite sets admit a bijection exactly when their cardinalities agree, giving No. This asks about existence rather than testing a given function; two empty sets admit the unique empty bijection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0077/)

Bijectivity and unique preimages · qf-bijective-cardinality-possibility · candidate





## 78. Existence of a Bijection from Finite Cardinalities

The explicit finite cardinalities are |A|=1 and |B|=1, with no particular function table supplied. Does some bijection A→B exist?

View answer and explanationYes

Finite sets admit a bijection exactly when their cardinalities agree, giving Yes. This asks about existence rather than testing a given function; two empty sets admit the unique empty bijection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0078/)

Bijectivity and unique preimages · qf-bijective-cardinality-possibility · candidate





## 79. Existence of a Bijection from Finite Cardinalities

The explicit finite cardinalities are |A|=1 and |B|=2, with no particular function table supplied. Does some bijection A→B exist?

View answer and explanationNo

Finite sets admit a bijection exactly when their cardinalities agree, giving No. This asks about existence rather than testing a given function; two empty sets admit the unique empty bijection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0079/)

Bijectivity and unique preimages · qf-bijective-cardinality-possibility · candidate





## 80. Existence of a Bijection from Finite Cardinalities

The explicit finite cardinalities are |A|=2 and |B|=2, with no particular function table supplied. Does some bijection A→B exist?

View answer and explanationYes

Finite sets admit a bijection exactly when their cardinalities agree, giving Yes. This asks about existence rather than testing a given function; two empty sets admit the unique empty bijection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0080/)

Bijectivity and unique preimages · qf-bijective-cardinality-possibility · candidate





## 81. Existence of a Bijection from Finite Cardinalities

The explicit finite cardinalities are |A|=2 and |B|=3, with no particular function table supplied. Does some bijection A→B exist?

View answer and explanationNo

Finite sets admit a bijection exactly when their cardinalities agree, giving No. This asks about existence rather than testing a given function; two empty sets admit the unique empty bijection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0081/)

Bijectivity and unique preimages · qf-bijective-cardinality-possibility · candidate





## 82. Existence of a Bijection from Finite Cardinalities

The explicit finite cardinalities are |A|=3 and |B|=3, with no particular function table supplied. Does some bijection A→B exist?

View answer and explanationYes

Finite sets admit a bijection exactly when their cardinalities agree, giving Yes. This asks about existence rather than testing a given function; two empty sets admit the unique empty bijection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0082/)

Bijectivity and unique preimages · qf-bijective-cardinality-possibility · candidate





## 83. Existence of a Bijection from Finite Cardinalities

The explicit finite cardinalities are |A|=3 and |B|=4, with no particular function table supplied. Does some bijection A→B exist?

View answer and explanationNo

Finite sets admit a bijection exactly when their cardinalities agree, giving No. This asks about existence rather than testing a given function; two empty sets admit the unique empty bijection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0083/)

Bijectivity and unique preimages · qf-bijective-cardinality-possibility · candidate





## 84. Existence of a Bijection from Finite Cardinalities

The explicit finite cardinalities are |A|=4 and |B|=4, with no particular function table supplied. Does some bijection A→B exist?

View answer and explanationYes

Finite sets admit a bijection exactly when their cardinalities agree, giving Yes. This asks about existence rather than testing a given function; two empty sets admit the unique empty bijection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0084/)

Bijectivity and unique preimages · qf-bijective-cardinality-possibility · candidate





## 85. Existence of a Bijection from Finite Cardinalities

The explicit finite cardinalities are |A|=4 and |B|=5, with no particular function table supplied. Does some bijection A→B exist?

View answer and explanationNo

Finite sets admit a bijection exactly when their cardinalities agree, giving No. This asks about existence rather than testing a given function; two empty sets admit the unique empty bijection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0085/)

Bijectivity and unique preimages · qf-bijective-cardinality-possibility · candidate





## 86. Existence of a Bijection from Finite Cardinalities

The explicit finite cardinalities are |A|=5 and |B|=5, with no particular function table supplied. Does some bijection A→B exist?

View answer and explanationYes

Finite sets admit a bijection exactly when their cardinalities agree, giving Yes. This asks about existence rather than testing a given function; two empty sets admit the unique empty bijection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0086/)

Bijectivity and unique preimages · qf-bijective-cardinality-possibility · candidate





## 87. Existence of a Bijection from Finite Cardinalities

The explicit finite cardinalities are |A|=5 and |B|=6, with no particular function table supplied. Does some bijection A→B exist?

View answer and explanationNo

Finite sets admit a bijection exactly when their cardinalities agree, giving No. This asks about existence rather than testing a given function; two empty sets admit the unique empty bijection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0087/)

Bijectivity and unique preimages · qf-bijective-cardinality-possibility · candidate





## 88. Existence of a Bijection from Finite Cardinalities

The explicit finite cardinalities are |A|=6 and |B|=6, with no particular function table supplied. Does some bijection A→B exist?

View answer and explanationYes

Finite sets admit a bijection exactly when their cardinalities agree, giving Yes. This asks about existence rather than testing a given function; two empty sets admit the unique empty bijection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0088/)

Bijectivity and unique preimages · qf-bijective-cardinality-possibility · candidate





## 89. Existence of a Bijection from Finite Cardinalities

The explicit finite cardinalities are |A|=6 and |B|=7, with no particular function table supplied. Does some bijection A→B exist?

View answer and explanationNo

Finite sets admit a bijection exactly when their cardinalities agree, giving No. This asks about existence rather than testing a given function; two empty sets admit the unique empty bijection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0089/)

Bijectivity and unique preimages · qf-bijective-cardinality-possibility · candidate





## 90. Existence of a Bijection from Finite Cardinalities

The explicit finite cardinalities are |A|=7 and |B|=7, with no particular function table supplied. Does some bijection A→B exist?

View answer and explanationYes

Finite sets admit a bijection exactly when their cardinalities agree, giving Yes. This asks about existence rather than testing a given function; two empty sets admit the unique empty bijection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0090/)

Bijectivity and unique preimages · qf-bijective-cardinality-possibility · candidate





## 91. Existence of a Bijection from Finite Cardinalities

The explicit finite cardinalities are |A|=7 and |B|=8, with no particular function table supplied. Does some bijection A→B exist?

View answer and explanationNo

Finite sets admit a bijection exactly when their cardinalities agree, giving No. This asks about existence rather than testing a given function; two empty sets admit the unique empty bijection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0091/)

Bijectivity and unique preimages · qf-bijective-cardinality-possibility · candidate





## 92. Existence of a Bijection from Finite Cardinalities

The explicit finite cardinalities are |A|=8 and |B|=8, with no particular function table supplied. Does some bijection A→B exist?

View answer and explanationYes

Finite sets admit a bijection exactly when their cardinalities agree, giving Yes. This asks about existence rather than testing a given function; two empty sets admit the unique empty bijection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0092/)

Bijectivity and unique preimages · qf-bijective-cardinality-possibility · candidate





## 93. Existence of a Bijection from Finite Cardinalities

The explicit finite cardinalities are |A|=8 and |B|=9, with no particular function table supplied. Does some bijection A→B exist?

View answer and explanationNo

Finite sets admit a bijection exactly when their cardinalities agree, giving No. This asks about existence rather than testing a given function; two empty sets admit the unique empty bijection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0093/)

Bijectivity and unique preimages · qf-bijective-cardinality-possibility · candidate





## 94. Existence of a Bijection from Finite Cardinalities

The explicit finite cardinalities are |A|=9 and |B|=9, with no particular function table supplied. Does some bijection A→B exist?

View answer and explanationYes

Finite sets admit a bijection exactly when their cardinalities agree, giving Yes. This asks about existence rather than testing a given function; two empty sets admit the unique empty bijection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0094/)

Bijectivity and unique preimages · qf-bijective-cardinality-possibility · candidate





## 95. Existence of a Bijection from Finite Cardinalities

The explicit finite cardinalities are |A|=9 and |B|=10, with no particular function table supplied. Does some bijection A→B exist?

View answer and explanationNo

Finite sets admit a bijection exactly when their cardinalities agree, giving No. This asks about existence rather than testing a given function; two empty sets admit the unique empty bijection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0095/)

Bijectivity and unique preimages · qf-bijective-cardinality-possibility · candidate





## 96. Existence of a Bijection from Finite Cardinalities

The explicit finite cardinalities are |A|=10 and |B|=10, with no particular function table supplied. Does some bijection A→B exist?

View answer and explanationYes

Finite sets admit a bijection exactly when their cardinalities agree, giving Yes. This asks about existence rather than testing a given function; two empty sets admit the unique empty bijection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0096/)

Bijectivity and unique preimages · qf-bijective-cardinality-possibility · candidate





## 97. Existence of a Bijection from Finite Cardinalities

The explicit finite cardinalities are |A|=10 and |B|=11, with no particular function table supplied. Does some bijection A→B exist?

View answer and explanationNo

Finite sets admit a bijection exactly when their cardinalities agree, giving No. This asks about existence rather than testing a given function; two empty sets admit the unique empty bijection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0097/)

Bijectivity and unique preimages · qf-bijective-cardinality-possibility · candidate





## 98. Existence of a Bijection from Finite Cardinalities

The explicit finite cardinalities are |A|=11 and |B|=11, with no particular function table supplied. Does some bijection A→B exist?

View answer and explanationYes

Finite sets admit a bijection exactly when their cardinalities agree, giving Yes. This asks about existence rather than testing a given function; two empty sets admit the unique empty bijection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0098/)

Bijectivity and unique preimages · qf-bijective-cardinality-possibility · candidate





## 99. Existence of a Bijection from Finite Cardinalities

The explicit finite cardinalities are |A|=11 and |B|=12, with no particular function table supplied. Does some bijection A→B exist?

View answer and explanationNo

Finite sets admit a bijection exactly when their cardinalities agree, giving No. This asks about existence rather than testing a given function; two empty sets admit the unique empty bijection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0099/)

Bijectivity and unique preimages · qf-bijective-cardinality-possibility · candidate





## 100. Existence of a Bijection from Finite Cardinalities

The explicit finite cardinalities are |A|=12 and |B|=12, with no particular function table supplied. Does some bijection A→B exist?

View answer and explanationYes

Finite sets admit a bijection exactly when their cardinalities agree, giving Yes. This asks about existence rather than testing a given function; two empty sets admit the unique empty bijection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261004-advance-20261005-r03-bijective-0100/)

Bijectivity and unique preimages · qf-bijective-cardinality-possibility · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261004-advance-20261005-r03-bijective-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261004-advance-20261005-r03-bijective-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261004-advance-20261005-r03-bijective-foundations/validation-report.json)