# Binary Relation 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-20261003-advance-20261004-r03-relation-foundations · 2026-10-03 · 100 questions

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



## 1. Ordered-Pair Membership in a Binary Relation

The binary relation has source A=∅, target B=∅, and R=∅⊆A×B. Does the ordered pair (0, 1) belong to R?

View answer and explanationNo

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0001/)

Binary relations and set projections · qf-relation-pair-membership · candidate





## 2. Ordered-Pair Membership in a Binary Relation

The binary relation has source A={0}, target B={1}, and R=∅⊆A×B. Does the ordered pair (1, 0) belong to R?

View answer and explanationNo

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0002/)

Binary relations and set projections · qf-relation-pair-membership · candidate





## 3. Ordered-Pair Membership in a Binary Relation

The binary relation has source A={-10, -9, -8}, target B={-6, -5, -4}, and R={(-10, -6), (-10, -5)}⊆A×B. Does the ordered pair (-8, -6) belong to R?

View answer and explanationNo

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0003/)

Binary relations and set projections · qf-relation-pair-membership · candidate





## 4. Ordered-Pair Membership in a Binary Relation

The binary relation has source A={-9, -8, -7}, target B={-5, -4, -3}, and R={(-9, -4), (-8, -4)}⊆A×B. Does the ordered pair (-9, -4) belong to R?

View answer and explanationYes

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives Yes.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0004/)

Binary relations and set projections · qf-relation-pair-membership · candidate





## 5. Ordered-Pair Membership in a Binary Relation

The binary relation has source A={-8, -7, -6}, target B={-4, -3, -2}, and R={(-8, -4), (-7, -3), (-6, -2)}⊆A×B. Does the ordered pair (-4, -8) belong to R?

View answer and explanationNo

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0005/)

Binary relations and set projections · qf-relation-pair-membership · candidate





## 6. Ordered-Pair Membership in a Binary Relation

The binary relation has source A={-7, -6, -5}, target B={-3, -2, -1}, and R={(-6, -3), (-5, -2)}⊆A×B. Does the ordered pair (-5, -3) belong to R?

View answer and explanationNo

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0006/)

Binary relations and set projections · qf-relation-pair-membership · candidate





## 7. Ordered-Pair Membership in a Binary Relation

The binary relation has source A={-6, -5, -4}, target B={-2, -1, 0}, and R=∅⊆A×B. Does the ordered pair (-6, -2) belong to R?

View answer and explanationNo

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0007/)

Binary relations and set projections · qf-relation-pair-membership · candidate





## 8. Ordered-Pair Membership in a Binary Relation

The binary relation has source A={-5, -4, -3}, target B={-1, 0, 1}, and R={(-5, -1)}⊆A×B. Does the ordered pair (-1, -5) belong to R?

View answer and explanationNo

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0008/)

Binary relations and set projections · qf-relation-pair-membership · candidate





## 9. Ordered-Pair Membership in a Binary Relation

The binary relation has source A={-4, -3, -2}, target B={0, 1, 2}, and R={(-4, 0), (-4, 1)}⊆A×B. Does the ordered pair (-2, 0) belong to R?

View answer and explanationNo

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0009/)

Binary relations and set projections · qf-relation-pair-membership · candidate





## 10. Ordered-Pair Membership in a Binary Relation

The binary relation has source A={-3, -2, -1}, target B={1, 2, 3}, and R={(-3, 2), (-2, 2)}⊆A×B. Does the ordered pair (-3, 2) belong to R?

View answer and explanationYes

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives Yes.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0010/)

Binary relations and set projections · qf-relation-pair-membership · candidate





## 11. Ordered-Pair Membership in a Binary Relation

The binary relation has source A={-2, -1, 0}, target B={2, 3, 4}, and R={(-2, 2), (-1, 3), (0, 4)}⊆A×B. Does the ordered pair (2, -2) belong to R?

View answer and explanationNo

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0011/)

Binary relations and set projections · qf-relation-pair-membership · candidate





## 12. Ordered-Pair Membership in a Binary Relation

The binary relation has source A={-1, 0, 1}, target B={3, 4, 5}, and R={(0, 3), (1, 4)}⊆A×B. Does the ordered pair (1, 3) belong to R?

View answer and explanationNo

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0012/)

Binary relations and set projections · qf-relation-pair-membership · candidate





## 13. Ordered-Pair Membership in a Binary Relation

The binary relation has source A={0, 1, 2}, target B={4, 5, 6}, and R=∅⊆A×B. Does the ordered pair (0, 4) belong to R?

View answer and explanationNo

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0013/)

Binary relations and set projections · qf-relation-pair-membership · candidate





## 14. Ordered-Pair Membership in a Binary Relation

The binary relation has source A={1, 2, 3}, target B={5, 6, 7}, and R={(1, 5)}⊆A×B. Does the ordered pair (5, 1) belong to R?

View answer and explanationNo

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0014/)

Binary relations and set projections · qf-relation-pair-membership · candidate





## 15. Ordered-Pair Membership in a Binary Relation

The binary relation has source A={2, 3, 4}, target B={6, 7, 8}, and R={(2, 6), (2, 7)}⊆A×B. Does the ordered pair (4, 6) belong to R?

View answer and explanationNo

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0015/)

Binary relations and set projections · qf-relation-pair-membership · candidate





## 16. Ordered-Pair Membership in a Binary Relation

The binary relation has source A={3, 4, 5}, target B={7, 8, 9}, and R={(3, 8), (4, 8)}⊆A×B. Does the ordered pair (3, 8) belong to R?

View answer and explanationYes

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives Yes.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0016/)

Binary relations and set projections · qf-relation-pair-membership · candidate





## 17. Ordered-Pair Membership in a Binary Relation

The binary relation has source A={4, 5, 6}, target B={8, 9, 10}, and R={(4, 8), (5, 9), (6, 10)}⊆A×B. Does the ordered pair (8, 4) belong to R?

View answer and explanationNo

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0017/)

Binary relations and set projections · qf-relation-pair-membership · candidate





## 18. Ordered-Pair Membership in a Binary Relation

The binary relation has source A={5, 6, 7}, target B={9, 10, 11}, and R={(6, 9), (7, 10)}⊆A×B. Does the ordered pair (7, 9) belong to R?

View answer and explanationNo

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0018/)

Binary relations and set projections · qf-relation-pair-membership · candidate





## 19. Ordered-Pair Membership in a Binary Relation

The binary relation has source A={6, 7, 8}, target B={10, 11, 12}, and R=∅⊆A×B. Does the ordered pair (6, 10) belong to R?

View answer and explanationNo

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0019/)

Binary relations and set projections · qf-relation-pair-membership · candidate





## 20. Ordered-Pair Membership in a Binary Relation

The binary relation has source A={7, 8, 9}, target B={11, 12, 13}, and R={(7, 11)}⊆A×B. Does the ordered pair (11, 7) belong to R?

View answer and explanationNo

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0020/)

Binary relations and set projections · qf-relation-pair-membership · candidate





## 21. Ordered-Pair Membership in a Binary Relation

The binary relation has source A={8, 9, 10}, target B={12, 13, 14}, and R={(8, 12), (8, 13)}⊆A×B. Does the ordered pair (10, 12) belong to R?

View answer and explanationNo

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0021/)

Binary relations and set projections · qf-relation-pair-membership · candidate





## 22. Ordered-Pair Membership in a Binary Relation

The binary relation has source A={9, 10, 11}, target B={13, 14, 15}, and R={(9, 14), (10, 14)}⊆A×B. Does the ordered pair (9, 14) belong to R?

View answer and explanationYes

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives Yes.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0022/)

Binary relations and set projections · qf-relation-pair-membership · candidate





## 23. Ordered-Pair Membership in a Binary Relation

The binary relation has source A={10, 11, 12}, target B={14, 15, 16}, and R={(10, 14), (11, 15), (12, 16)}⊆A×B. Does the ordered pair (14, 10) belong to R?

View answer and explanationNo

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0023/)

Binary relations and set projections · qf-relation-pair-membership · candidate





## 24. Ordered-Pair Membership in a Binary Relation

The binary relation has source A={11, 12, 13}, target B={15, 16, 17}, and R={(12, 15), (13, 16)}⊆A×B. Does the ordered pair (13, 15) belong to R?

View answer and explanationNo

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0024/)

Binary relations and set projections · qf-relation-pair-membership · candidate





## 25. Ordered-Pair Membership in a Binary Relation

The binary relation has source A={12, 13, 14}, target B={16, 17, 18}, and R=∅⊆A×B. Does the ordered pair (12, 16) belong to R?

View answer and explanationNo

Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0025/)

Binary relations and set projections · qf-relation-pair-membership · candidate





## 26. First Projection of a Binary Relation

The binary relation has source A=∅, target B=∅, and R=∅⊆A×B. List its actual domain Dom(R) in increasing order with each element once.

View answer and explanation∅

Collect only first components that actually occur in R. Dom(R)=∅; the declared source A need not equal this projection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0026/)

Binary relations and set projections · qf-relation-domain-projection · candidate





## 27. First Projection of a Binary Relation

The binary relation has source A={0}, target B={1}, and R=∅⊆A×B. List its actual domain Dom(R) in increasing order with each element once.

View answer and explanation∅

Collect only first components that actually occur in R. Dom(R)=∅; the declared source A need not equal this projection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0027/)

Binary relations and set projections · qf-relation-domain-projection · candidate





## 28. First Projection of a Binary Relation

The binary relation has source A={-10, -9, -8}, target B={-6, -5, -4}, and R={(-10, -6), (-10, -5)}⊆A×B. List its actual domain Dom(R) in increasing order with each element once.

View answer and explanation{-10}

Collect only first components that actually occur in R. Dom(R)={-10}; the declared source A need not equal this projection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0028/)

Binary relations and set projections · qf-relation-domain-projection · candidate





## 29. First Projection of a Binary Relation

The binary relation has source A={-9, -8, -7}, target B={-5, -4, -3}, and R={(-9, -4), (-8, -4)}⊆A×B. List its actual domain Dom(R) in increasing order with each element once.

View answer and explanation{-9, -8}

Collect only first components that actually occur in R. Dom(R)={-9, -8}; the declared source A need not equal this projection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0029/)

Binary relations and set projections · qf-relation-domain-projection · candidate





## 30. First Projection of a Binary Relation

The binary relation has source A={-8, -7, -6}, target B={-4, -3, -2}, and R={(-8, -4), (-7, -3), (-6, -2)}⊆A×B. List its actual domain Dom(R) in increasing order with each element once.

View answer and explanation{-8, -7, -6}

Collect only first components that actually occur in R. Dom(R)={-8, -7, -6}; the declared source A need not equal this projection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0030/)

Binary relations and set projections · qf-relation-domain-projection · candidate





## 31. First Projection of a Binary Relation

The binary relation has source A={-7, -6, -5}, target B={-3, -2, -1}, and R={(-6, -3), (-5, -2)}⊆A×B. List its actual domain Dom(R) in increasing order with each element once.

View answer and explanation{-6, -5}

Collect only first components that actually occur in R. Dom(R)={-6, -5}; the declared source A need not equal this projection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0031/)

Binary relations and set projections · qf-relation-domain-projection · candidate





## 32. First Projection of a Binary Relation

The binary relation has source A={-6, -5, -4}, target B={-2, -1, 0}, and R=∅⊆A×B. List its actual domain Dom(R) in increasing order with each element once.

View answer and explanation∅

Collect only first components that actually occur in R. Dom(R)=∅; the declared source A need not equal this projection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0032/)

Binary relations and set projections · qf-relation-domain-projection · candidate





## 33. First Projection of a Binary Relation

The binary relation has source A={-5, -4, -3}, target B={-1, 0, 1}, and R={(-5, -1)}⊆A×B. List its actual domain Dom(R) in increasing order with each element once.

View answer and explanation{-5}

Collect only first components that actually occur in R. Dom(R)={-5}; the declared source A need not equal this projection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0033/)

Binary relations and set projections · qf-relation-domain-projection · candidate





## 34. First Projection of a Binary Relation

The binary relation has source A={-4, -3, -2}, target B={0, 1, 2}, and R={(-4, 0), (-4, 1)}⊆A×B. List its actual domain Dom(R) in increasing order with each element once.

View answer and explanation{-4}

Collect only first components that actually occur in R. Dom(R)={-4}; the declared source A need not equal this projection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0034/)

Binary relations and set projections · qf-relation-domain-projection · candidate





## 35. First Projection of a Binary Relation

The binary relation has source A={-3, -2, -1}, target B={1, 2, 3}, and R={(-3, 2), (-2, 2)}⊆A×B. List its actual domain Dom(R) in increasing order with each element once.

View answer and explanation{-3, -2}

Collect only first components that actually occur in R. Dom(R)={-3, -2}; the declared source A need not equal this projection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0035/)

Binary relations and set projections · qf-relation-domain-projection · candidate





## 36. First Projection of a Binary Relation

The binary relation has source A={-2, -1, 0}, target B={2, 3, 4}, and R={(-2, 2), (-1, 3), (0, 4)}⊆A×B. List its actual domain Dom(R) in increasing order with each element once.

View answer and explanation{-2, -1, 0}

Collect only first components that actually occur in R. Dom(R)={-2, -1, 0}; the declared source A need not equal this projection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0036/)

Binary relations and set projections · qf-relation-domain-projection · candidate





## 37. First Projection of a Binary Relation

The binary relation has source A={-1, 0, 1}, target B={3, 4, 5}, and R={(0, 3), (1, 4)}⊆A×B. List its actual domain Dom(R) in increasing order with each element once.

View answer and explanation{0, 1}

Collect only first components that actually occur in R. Dom(R)={0, 1}; the declared source A need not equal this projection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0037/)

Binary relations and set projections · qf-relation-domain-projection · candidate





## 38. First Projection of a Binary Relation

The binary relation has source A={0, 1, 2}, target B={4, 5, 6}, and R=∅⊆A×B. List its actual domain Dom(R) in increasing order with each element once.

View answer and explanation∅

Collect only first components that actually occur in R. Dom(R)=∅; the declared source A need not equal this projection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0038/)

Binary relations and set projections · qf-relation-domain-projection · candidate





## 39. First Projection of a Binary Relation

The binary relation has source A={1, 2, 3}, target B={5, 6, 7}, and R={(1, 5)}⊆A×B. List its actual domain Dom(R) in increasing order with each element once.

View answer and explanation{1}

Collect only first components that actually occur in R. Dom(R)={1}; the declared source A need not equal this projection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0039/)

Binary relations and set projections · qf-relation-domain-projection · candidate





## 40. First Projection of a Binary Relation

The binary relation has source A={2, 3, 4}, target B={6, 7, 8}, and R={(2, 6), (2, 7)}⊆A×B. List its actual domain Dom(R) in increasing order with each element once.

View answer and explanation{2}

Collect only first components that actually occur in R. Dom(R)={2}; the declared source A need not equal this projection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0040/)

Binary relations and set projections · qf-relation-domain-projection · candidate





## 41. First Projection of a Binary Relation

The binary relation has source A={3, 4, 5}, target B={7, 8, 9}, and R={(3, 8), (4, 8)}⊆A×B. List its actual domain Dom(R) in increasing order with each element once.

View answer and explanation{3, 4}

Collect only first components that actually occur in R. Dom(R)={3, 4}; the declared source A need not equal this projection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0041/)

Binary relations and set projections · qf-relation-domain-projection · candidate





## 42. First Projection of a Binary Relation

The binary relation has source A={4, 5, 6}, target B={8, 9, 10}, and R={(4, 8), (5, 9), (6, 10)}⊆A×B. List its actual domain Dom(R) in increasing order with each element once.

View answer and explanation{4, 5, 6}

Collect only first components that actually occur in R. Dom(R)={4, 5, 6}; the declared source A need not equal this projection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0042/)

Binary relations and set projections · qf-relation-domain-projection · candidate





## 43. First Projection of a Binary Relation

The binary relation has source A={5, 6, 7}, target B={9, 10, 11}, and R={(6, 9), (7, 10)}⊆A×B. List its actual domain Dom(R) in increasing order with each element once.

View answer and explanation{6, 7}

Collect only first components that actually occur in R. Dom(R)={6, 7}; the declared source A need not equal this projection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0043/)

Binary relations and set projections · qf-relation-domain-projection · candidate





## 44. First Projection of a Binary Relation

The binary relation has source A={6, 7, 8}, target B={10, 11, 12}, and R=∅⊆A×B. List its actual domain Dom(R) in increasing order with each element once.

View answer and explanation∅

Collect only first components that actually occur in R. Dom(R)=∅; the declared source A need not equal this projection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0044/)

Binary relations and set projections · qf-relation-domain-projection · candidate





## 45. First Projection of a Binary Relation

The binary relation has source A={7, 8, 9}, target B={11, 12, 13}, and R={(7, 11)}⊆A×B. List its actual domain Dom(R) in increasing order with each element once.

View answer and explanation{7}

Collect only first components that actually occur in R. Dom(R)={7}; the declared source A need not equal this projection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0045/)

Binary relations and set projections · qf-relation-domain-projection · candidate





## 46. First Projection of a Binary Relation

The binary relation has source A={8, 9, 10}, target B={12, 13, 14}, and R={(8, 12), (8, 13)}⊆A×B. List its actual domain Dom(R) in increasing order with each element once.

View answer and explanation{8}

Collect only first components that actually occur in R. Dom(R)={8}; the declared source A need not equal this projection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0046/)

Binary relations and set projections · qf-relation-domain-projection · candidate





## 47. First Projection of a Binary Relation

The binary relation has source A={9, 10, 11}, target B={13, 14, 15}, and R={(9, 14), (10, 14)}⊆A×B. List its actual domain Dom(R) in increasing order with each element once.

View answer and explanation{9, 10}

Collect only first components that actually occur in R. Dom(R)={9, 10}; the declared source A need not equal this projection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0047/)

Binary relations and set projections · qf-relation-domain-projection · candidate





## 48. First Projection of a Binary Relation

The binary relation has source A={10, 11, 12}, target B={14, 15, 16}, and R={(10, 14), (11, 15), (12, 16)}⊆A×B. List its actual domain Dom(R) in increasing order with each element once.

View answer and explanation{10, 11, 12}

Collect only first components that actually occur in R. Dom(R)={10, 11, 12}; the declared source A need not equal this projection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0048/)

Binary relations and set projections · qf-relation-domain-projection · candidate





## 49. First Projection of a Binary Relation

The binary relation has source A={11, 12, 13}, target B={15, 16, 17}, and R={(12, 15), (13, 16)}⊆A×B. List its actual domain Dom(R) in increasing order with each element once.

View answer and explanation{12, 13}

Collect only first components that actually occur in R. Dom(R)={12, 13}; the declared source A need not equal this projection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0049/)

Binary relations and set projections · qf-relation-domain-projection · candidate





## 50. First Projection of a Binary Relation

The binary relation has source A={12, 13, 14}, target B={16, 17, 18}, and R=∅⊆A×B. List its actual domain Dom(R) in increasing order with each element once.

View answer and explanation∅

Collect only first components that actually occur in R. Dom(R)=∅; the declared source A need not equal this projection.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0050/)

Binary relations and set projections · qf-relation-domain-projection · candidate





## 51. Image of a Subset under a Binary Relation

The binary relation has source A=∅, target B=∅, and R=∅⊆A×B. For the explicit C=∅⊆A, list R[C] in increasing order with no duplicates.

View answer and explanation∅

Select relation pairs whose first component lies in C and collect their second components. R[C]=∅; a relation need not have a unique output for each input.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0051/)

Binary relations and set projections · qf-relation-subset-image · candidate





## 52. Image of a Subset under a Binary Relation

The binary relation has source A={0}, target B={1}, and R=∅⊆A×B. For the explicit C={0}⊆A, list R[C] in increasing order with no duplicates.

View answer and explanation∅

Select relation pairs whose first component lies in C and collect their second components. R[C]=∅; a relation need not have a unique output for each input.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0052/)

Binary relations and set projections · qf-relation-subset-image · candidate





## 53. Image of a Subset under a Binary Relation

The binary relation has source A={-10, -9, -8}, target B={-6, -5, -4}, and R={(-10, -6), (-10, -5)}⊆A×B. For the explicit C={-10}⊆A, list R[C] in increasing order with no duplicates.

View answer and explanation{-6, -5}

Select relation pairs whose first component lies in C and collect their second components. R[C]={-6, -5}; a relation need not have a unique output for each input.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0053/)

Binary relations and set projections · qf-relation-subset-image · candidate





## 54. Image of a Subset under a Binary Relation

The binary relation has source A={-9, -8, -7}, target B={-5, -4, -3}, and R={(-9, -4), (-8, -4)}⊆A×B. For the explicit C={-8, -7}⊆A, list R[C] in increasing order with no duplicates.

View answer and explanation{-4}

Select relation pairs whose first component lies in C and collect their second components. R[C]={-4}; a relation need not have a unique output for each input.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0054/)

Binary relations and set projections · qf-relation-subset-image · candidate





## 55. Image of a Subset under a Binary Relation

The binary relation has source A={-8, -7, -6}, target B={-4, -3, -2}, and R={(-8, -4), (-7, -3), (-6, -2)}⊆A×B. For the explicit C=∅⊆A, list R[C] in increasing order with no duplicates.

View answer and explanation∅

Select relation pairs whose first component lies in C and collect their second components. R[C]=∅; a relation need not have a unique output for each input.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0055/)

Binary relations and set projections · qf-relation-subset-image · candidate





## 56. Image of a Subset under a Binary Relation

The binary relation has source A={-7, -6, -5}, target B={-3, -2, -1}, and R={(-6, -3), (-5, -2)}⊆A×B. For the explicit C={-7, -6, -5}⊆A, list R[C] in increasing order with no duplicates.

View answer and explanation{-3, -2}

Select relation pairs whose first component lies in C and collect their second components. R[C]={-3, -2}; a relation need not have a unique output for each input.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0056/)

Binary relations and set projections · qf-relation-subset-image · candidate





## 57. Image of a Subset under a Binary Relation

The binary relation has source A={-6, -5, -4}, target B={-2, -1, 0}, and R=∅⊆A×B. For the explicit C={-6}⊆A, list R[C] in increasing order with no duplicates.

View answer and explanation∅

Select relation pairs whose first component lies in C and collect their second components. R[C]=∅; a relation need not have a unique output for each input.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0057/)

Binary relations and set projections · qf-relation-subset-image · candidate





## 58. Image of a Subset under a Binary Relation

The binary relation has source A={-5, -4, -3}, target B={-1, 0, 1}, and R={(-5, -1)}⊆A×B. For the explicit C={-4, -3}⊆A, list R[C] in increasing order with no duplicates.

View answer and explanation∅

Select relation pairs whose first component lies in C and collect their second components. R[C]=∅; a relation need not have a unique output for each input.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0058/)

Binary relations and set projections · qf-relation-subset-image · candidate





## 59. Image of a Subset under a Binary Relation

The binary relation has source A={-4, -3, -2}, target B={0, 1, 2}, and R={(-4, 0), (-4, 1)}⊆A×B. For the explicit C=∅⊆A, list R[C] in increasing order with no duplicates.

View answer and explanation∅

Select relation pairs whose first component lies in C and collect their second components. R[C]=∅; a relation need not have a unique output for each input.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0059/)

Binary relations and set projections · qf-relation-subset-image · candidate





## 60. Image of a Subset under a Binary Relation

The binary relation has source A={-3, -2, -1}, target B={1, 2, 3}, and R={(-3, 2), (-2, 2)}⊆A×B. For the explicit C={-3, -2, -1}⊆A, list R[C] in increasing order with no duplicates.

View answer and explanation{2}

Select relation pairs whose first component lies in C and collect their second components. R[C]={2}; a relation need not have a unique output for each input.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0060/)

Binary relations and set projections · qf-relation-subset-image · candidate





## 61. Image of a Subset under a Binary Relation

The binary relation has source A={-2, -1, 0}, target B={2, 3, 4}, and R={(-2, 2), (-1, 3), (0, 4)}⊆A×B. For the explicit C={-2}⊆A, list R[C] in increasing order with no duplicates.

View answer and explanation{2}

Select relation pairs whose first component lies in C and collect their second components. R[C]={2}; a relation need not have a unique output for each input.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0061/)

Binary relations and set projections · qf-relation-subset-image · candidate





## 62. Image of a Subset under a Binary Relation

The binary relation has source A={-1, 0, 1}, target B={3, 4, 5}, and R={(0, 3), (1, 4)}⊆A×B. For the explicit C={0, 1}⊆A, list R[C] in increasing order with no duplicates.

View answer and explanation{3, 4}

Select relation pairs whose first component lies in C and collect their second components. R[C]={3, 4}; a relation need not have a unique output for each input.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0062/)

Binary relations and set projections · qf-relation-subset-image · candidate





## 63. Image of a Subset under a Binary Relation

The binary relation has source A={0, 1, 2}, target B={4, 5, 6}, and R=∅⊆A×B. For the explicit C=∅⊆A, list R[C] in increasing order with no duplicates.

View answer and explanation∅

Select relation pairs whose first component lies in C and collect their second components. R[C]=∅; a relation need not have a unique output for each input.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0063/)

Binary relations and set projections · qf-relation-subset-image · candidate





## 64. Image of a Subset under a Binary Relation

The binary relation has source A={1, 2, 3}, target B={5, 6, 7}, and R={(1, 5)}⊆A×B. For the explicit C={1, 2, 3}⊆A, list R[C] in increasing order with no duplicates.

View answer and explanation{5}

Select relation pairs whose first component lies in C and collect their second components. R[C]={5}; a relation need not have a unique output for each input.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0064/)

Binary relations and set projections · qf-relation-subset-image · candidate





## 65. Image of a Subset under a Binary Relation

The binary relation has source A={2, 3, 4}, target B={6, 7, 8}, and R={(2, 6), (2, 7)}⊆A×B. For the explicit C={2}⊆A, list R[C] in increasing order with no duplicates.

View answer and explanation{6, 7}

Select relation pairs whose first component lies in C and collect their second components. R[C]={6, 7}; a relation need not have a unique output for each input.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0065/)

Binary relations and set projections · qf-relation-subset-image · candidate





## 66. Image of a Subset under a Binary Relation

The binary relation has source A={3, 4, 5}, target B={7, 8, 9}, and R={(3, 8), (4, 8)}⊆A×B. For the explicit C={4, 5}⊆A, list R[C] in increasing order with no duplicates.

View answer and explanation{8}

Select relation pairs whose first component lies in C and collect their second components. R[C]={8}; a relation need not have a unique output for each input.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0066/)

Binary relations and set projections · qf-relation-subset-image · candidate





## 67. Image of a Subset under a Binary Relation

The binary relation has source A={4, 5, 6}, target B={8, 9, 10}, and R={(4, 8), (5, 9), (6, 10)}⊆A×B. For the explicit C=∅⊆A, list R[C] in increasing order with no duplicates.

View answer and explanation∅

Select relation pairs whose first component lies in C and collect their second components. R[C]=∅; a relation need not have a unique output for each input.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0067/)

Binary relations and set projections · qf-relation-subset-image · candidate





## 68. Image of a Subset under a Binary Relation

The binary relation has source A={5, 6, 7}, target B={9, 10, 11}, and R={(6, 9), (7, 10)}⊆A×B. For the explicit C={5, 6, 7}⊆A, list R[C] in increasing order with no duplicates.

View answer and explanation{9, 10}

Select relation pairs whose first component lies in C and collect their second components. R[C]={9, 10}; a relation need not have a unique output for each input.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0068/)

Binary relations and set projections · qf-relation-subset-image · candidate





## 69. Image of a Subset under a Binary Relation

The binary relation has source A={6, 7, 8}, target B={10, 11, 12}, and R=∅⊆A×B. For the explicit C={6}⊆A, list R[C] in increasing order with no duplicates.

View answer and explanation∅

Select relation pairs whose first component lies in C and collect their second components. R[C]=∅; a relation need not have a unique output for each input.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0069/)

Binary relations and set projections · qf-relation-subset-image · candidate





## 70. Image of a Subset under a Binary Relation

The binary relation has source A={7, 8, 9}, target B={11, 12, 13}, and R={(7, 11)}⊆A×B. For the explicit C={8, 9}⊆A, list R[C] in increasing order with no duplicates.

View answer and explanation∅

Select relation pairs whose first component lies in C and collect their second components. R[C]=∅; a relation need not have a unique output for each input.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0070/)

Binary relations and set projections · qf-relation-subset-image · candidate





## 71. Image of a Subset under a Binary Relation

The binary relation has source A={8, 9, 10}, target B={12, 13, 14}, and R={(8, 12), (8, 13)}⊆A×B. For the explicit C=∅⊆A, list R[C] in increasing order with no duplicates.

View answer and explanation∅

Select relation pairs whose first component lies in C and collect their second components. R[C]=∅; a relation need not have a unique output for each input.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0071/)

Binary relations and set projections · qf-relation-subset-image · candidate





## 72. Image of a Subset under a Binary Relation

The binary relation has source A={9, 10, 11}, target B={13, 14, 15}, and R={(9, 14), (10, 14)}⊆A×B. For the explicit C={9, 10, 11}⊆A, list R[C] in increasing order with no duplicates.

View answer and explanation{14}

Select relation pairs whose first component lies in C and collect their second components. R[C]={14}; a relation need not have a unique output for each input.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0072/)

Binary relations and set projections · qf-relation-subset-image · candidate





## 73. Image of a Subset under a Binary Relation

The binary relation has source A={10, 11, 12}, target B={14, 15, 16}, and R={(10, 14), (11, 15), (12, 16)}⊆A×B. For the explicit C={10}⊆A, list R[C] in increasing order with no duplicates.

View answer and explanation{14}

Select relation pairs whose first component lies in C and collect their second components. R[C]={14}; a relation need not have a unique output for each input.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0073/)

Binary relations and set projections · qf-relation-subset-image · candidate





## 74. Image of a Subset under a Binary Relation

The binary relation has source A={11, 12, 13}, target B={15, 16, 17}, and R={(12, 15), (13, 16)}⊆A×B. For the explicit C={12, 13}⊆A, list R[C] in increasing order with no duplicates.

View answer and explanation{15, 16}

Select relation pairs whose first component lies in C and collect their second components. R[C]={15, 16}; a relation need not have a unique output for each input.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0074/)

Binary relations and set projections · qf-relation-subset-image · candidate





## 75. Image of a Subset under a Binary Relation

The binary relation has source A={12, 13, 14}, target B={16, 17, 18}, and R=∅⊆A×B. For the explicit C=∅⊆A, list R[C] in increasing order with no duplicates.

View answer and explanation∅

Select relation pairs whose first component lies in C and collect their second components. R[C]=∅; a relation need not have a unique output for each input.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0075/)

Binary relations and set projections · qf-relation-subset-image · candidate





## 76. Converse of a Binary Relation

The binary relation has source A=∅, target B=∅, and R=∅⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.

View answer and explanation∅

Exchange every (x,y) for (y,x), giving R⁻¹=∅. A converse relation is not automatically an inverse function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0076/)

Binary relations and set projections · qf-relation-converse-set · candidate





## 77. Converse of a Binary Relation

The binary relation has source A={0}, target B={1}, and R=∅⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.

View answer and explanation∅

Exchange every (x,y) for (y,x), giving R⁻¹=∅. A converse relation is not automatically an inverse function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0077/)

Binary relations and set projections · qf-relation-converse-set · candidate





## 78. Converse of a Binary Relation

The binary relation has source A={-10, -9, -8}, target B={-6, -5, -4}, and R={(-10, -6), (-10, -5)}⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.

View answer and explanation{(-6, -10), (-5, -10)}

Exchange every (x,y) for (y,x), giving R⁻¹={(-6, -10), (-5, -10)}. A converse relation is not automatically an inverse function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0078/)

Binary relations and set projections · qf-relation-converse-set · candidate





## 79. Converse of a Binary Relation

The binary relation has source A={-9, -8, -7}, target B={-5, -4, -3}, and R={(-9, -4), (-8, -4)}⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.

View answer and explanation{(-4, -9), (-4, -8)}

Exchange every (x,y) for (y,x), giving R⁻¹={(-4, -9), (-4, -8)}. A converse relation is not automatically an inverse function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0079/)

Binary relations and set projections · qf-relation-converse-set · candidate





## 80. Converse of a Binary Relation

The binary relation has source A={-8, -7, -6}, target B={-4, -3, -2}, and R={(-8, -4), (-7, -3), (-6, -2)}⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.

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

Exchange every (x,y) for (y,x), giving R⁻¹={(-4, -8), (-3, -7), (-2, -6)}. A converse relation is not automatically an inverse function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0080/)

Binary relations and set projections · qf-relation-converse-set · candidate





## 81. Converse of a Binary Relation

The binary relation has source A={-7, -6, -5}, target B={-3, -2, -1}, and R={(-6, -3), (-5, -2)}⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.

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

Exchange every (x,y) for (y,x), giving R⁻¹={(-3, -6), (-2, -5)}. A converse relation is not automatically an inverse function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0081/)

Binary relations and set projections · qf-relation-converse-set · candidate





## 82. Converse of a Binary Relation

The binary relation has source A={-6, -5, -4}, target B={-2, -1, 0}, and R=∅⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.

View answer and explanation∅

Exchange every (x,y) for (y,x), giving R⁻¹=∅. A converse relation is not automatically an inverse function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0082/)

Binary relations and set projections · qf-relation-converse-set · candidate





## 83. Converse of a Binary Relation

The binary relation has source A={-5, -4, -3}, target B={-1, 0, 1}, and R={(-5, -1)}⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.

View answer and explanation{(-1, -5)}

Exchange every (x,y) for (y,x), giving R⁻¹={(-1, -5)}. A converse relation is not automatically an inverse function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0083/)

Binary relations and set projections · qf-relation-converse-set · candidate





## 84. Converse of a Binary Relation

The binary relation has source A={-4, -3, -2}, target B={0, 1, 2}, and R={(-4, 0), (-4, 1)}⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.

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

Exchange every (x,y) for (y,x), giving R⁻¹={(0, -4), (1, -4)}. A converse relation is not automatically an inverse function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0084/)

Binary relations and set projections · qf-relation-converse-set · candidate





## 85. Converse of a Binary Relation

The binary relation has source A={-3, -2, -1}, target B={1, 2, 3}, and R={(-3, 2), (-2, 2)}⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.

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

Exchange every (x,y) for (y,x), giving R⁻¹={(2, -3), (2, -2)}. A converse relation is not automatically an inverse function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0085/)

Binary relations and set projections · qf-relation-converse-set · candidate





## 86. Converse of a Binary Relation

The binary relation has source A={-2, -1, 0}, target B={2, 3, 4}, and R={(-2, 2), (-1, 3), (0, 4)}⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.

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

Exchange every (x,y) for (y,x), giving R⁻¹={(2, -2), (3, -1), (4, 0)}. A converse relation is not automatically an inverse function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0086/)

Binary relations and set projections · qf-relation-converse-set · candidate





## 87. Converse of a Binary Relation

The binary relation has source A={-1, 0, 1}, target B={3, 4, 5}, and R={(0, 3), (1, 4)}⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.

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

Exchange every (x,y) for (y,x), giving R⁻¹={(3, 0), (4, 1)}. A converse relation is not automatically an inverse function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0087/)

Binary relations and set projections · qf-relation-converse-set · candidate





## 88. Converse of a Binary Relation

The binary relation has source A={0, 1, 2}, target B={4, 5, 6}, and R=∅⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.

View answer and explanation∅

Exchange every (x,y) for (y,x), giving R⁻¹=∅. A converse relation is not automatically an inverse function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0088/)

Binary relations and set projections · qf-relation-converse-set · candidate





## 89. Converse of a Binary Relation

The binary relation has source A={1, 2, 3}, target B={5, 6, 7}, and R={(1, 5)}⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.

View answer and explanation{(5, 1)}

Exchange every (x,y) for (y,x), giving R⁻¹={(5, 1)}. A converse relation is not automatically an inverse function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0089/)

Binary relations and set projections · qf-relation-converse-set · candidate





## 90. Converse of a Binary Relation

The binary relation has source A={2, 3, 4}, target B={6, 7, 8}, and R={(2, 6), (2, 7)}⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.

View answer and explanation{(6, 2), (7, 2)}

Exchange every (x,y) for (y,x), giving R⁻¹={(6, 2), (7, 2)}. A converse relation is not automatically an inverse function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0090/)

Binary relations and set projections · qf-relation-converse-set · candidate





## 91. Converse of a Binary Relation

The binary relation has source A={3, 4, 5}, target B={7, 8, 9}, and R={(3, 8), (4, 8)}⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.

View answer and explanation{(8, 3), (8, 4)}

Exchange every (x,y) for (y,x), giving R⁻¹={(8, 3), (8, 4)}. A converse relation is not automatically an inverse function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0091/)

Binary relations and set projections · qf-relation-converse-set · candidate





## 92. Converse of a Binary Relation

The binary relation has source A={4, 5, 6}, target B={8, 9, 10}, and R={(4, 8), (5, 9), (6, 10)}⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.

View answer and explanation{(8, 4), (9, 5), (10, 6)}

Exchange every (x,y) for (y,x), giving R⁻¹={(8, 4), (9, 5), (10, 6)}. A converse relation is not automatically an inverse function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0092/)

Binary relations and set projections · qf-relation-converse-set · candidate





## 93. Converse of a Binary Relation

The binary relation has source A={5, 6, 7}, target B={9, 10, 11}, and R={(6, 9), (7, 10)}⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.

View answer and explanation{(9, 6), (10, 7)}

Exchange every (x,y) for (y,x), giving R⁻¹={(9, 6), (10, 7)}. A converse relation is not automatically an inverse function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0093/)

Binary relations and set projections · qf-relation-converse-set · candidate





## 94. Converse of a Binary Relation

The binary relation has source A={6, 7, 8}, target B={10, 11, 12}, and R=∅⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.

View answer and explanation∅

Exchange every (x,y) for (y,x), giving R⁻¹=∅. A converse relation is not automatically an inverse function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0094/)

Binary relations and set projections · qf-relation-converse-set · candidate





## 95. Converse of a Binary Relation

The binary relation has source A={7, 8, 9}, target B={11, 12, 13}, and R={(7, 11)}⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.

View answer and explanation{(11, 7)}

Exchange every (x,y) for (y,x), giving R⁻¹={(11, 7)}. A converse relation is not automatically an inverse function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0095/)

Binary relations and set projections · qf-relation-converse-set · candidate





## 96. Converse of a Binary Relation

The binary relation has source A={8, 9, 10}, target B={12, 13, 14}, and R={(8, 12), (8, 13)}⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.

View answer and explanation{(12, 8), (13, 8)}

Exchange every (x,y) for (y,x), giving R⁻¹={(12, 8), (13, 8)}. A converse relation is not automatically an inverse function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0096/)

Binary relations and set projections · qf-relation-converse-set · candidate





## 97. Converse of a Binary Relation

The binary relation has source A={9, 10, 11}, target B={13, 14, 15}, and R={(9, 14), (10, 14)}⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.

View answer and explanation{(14, 9), (14, 10)}

Exchange every (x,y) for (y,x), giving R⁻¹={(14, 9), (14, 10)}. A converse relation is not automatically an inverse function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0097/)

Binary relations and set projections · qf-relation-converse-set · candidate





## 98. Converse of a Binary Relation

The binary relation has source A={10, 11, 12}, target B={14, 15, 16}, and R={(10, 14), (11, 15), (12, 16)}⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.

View answer and explanation{(14, 10), (15, 11), (16, 12)}

Exchange every (x,y) for (y,x), giving R⁻¹={(14, 10), (15, 11), (16, 12)}. A converse relation is not automatically an inverse function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0098/)

Binary relations and set projections · qf-relation-converse-set · candidate





## 99. Converse of a Binary Relation

The binary relation has source A={11, 12, 13}, target B={15, 16, 17}, and R={(12, 15), (13, 16)}⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.

View answer and explanation{(15, 12), (16, 13)}

Exchange every (x,y) for (y,x), giving R⁻¹={(15, 12), (16, 13)}. A converse relation is not automatically an inverse function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0099/)

Binary relations and set projections · qf-relation-converse-set · candidate





## 100. Converse of a Binary Relation

The binary relation has source A={12, 13, 14}, target B={16, 17, 18}, and R=∅⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.

View answer and explanation∅

Exchange every (x,y) for (y,x), giving R⁻¹=∅. A converse relation is not automatically an inverse function.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r03-relation-0100/)

Binary relations and set projections · qf-relation-converse-set · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261003-advance-20261004-r03-relation-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261003-advance-20261004-r03-relation-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261003-advance-20261004-r03-relation-foundations/validation-report.json)