# Foundations of Mathematical Proof

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-20260921-proof-foundations · 2026-09-21 · 100 questions

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



## 1. Modus Ponens

Let P mean “the integer n is divisible by 4” and Q mean “the integer n is even”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and P, exactly the form of modus ponens, so Q follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-ponens-0001/)

Mathematical proof · qf-proof-modus-ponens · candidate





## 2. Modus Ponens

Let P mean “the integer n is divisible by 6” and Q mean “the integer n is divisible by 3”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and P, exactly the form of modus ponens, so Q follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-ponens-0002/)

Mathematical proof · qf-proof-modus-ponens · candidate





## 3. Modus Ponens

Let P mean “the integer n is a multiple of 9” and Q mean “the integer n is a multiple of 3”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and P, exactly the form of modus ponens, so Q follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-ponens-0003/)

Mathematical proof · qf-proof-modus-ponens · candidate





## 4. Modus Ponens

Let P mean “the integer n is prime and greater than 2” and Q mean “the integer n is odd”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and P, exactly the form of modus ponens, so Q follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-ponens-0004/)

Mathematical proof · qf-proof-modus-ponens · candidate





## 5. Modus Ponens

Let P mean “the triangle has three equal sides” and Q mean “the triangle has at least two equal sides”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and P, exactly the form of modus ponens, so Q follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-ponens-0005/)

Mathematical proof · qf-proof-modus-ponens · candidate





## 6. Modus Ponens

Let P mean “the quadrilateral is a square” and Q mean “the quadrilateral has four right angles”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and P, exactly the form of modus ponens, so Q follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-ponens-0006/)

Mathematical proof · qf-proof-modus-ponens · candidate





## 7. Modus Ponens

Let P mean “the integer n is a multiple of 10” and Q mean “the integer n is a multiple of 5”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and P, exactly the form of modus ponens, so Q follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-ponens-0007/)

Mathematical proof · qf-proof-modus-ponens · candidate





## 8. Modus Ponens

Let P mean “the integer n is a multiple of 12” and Q mean “the integer n is a multiple of 4”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and P, exactly the form of modus ponens, so Q follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-ponens-0008/)

Mathematical proof · qf-proof-modus-ponens · candidate





## 9. Modus Ponens

Let P mean “the integer n is a perfect square” and Q mean “there exists an integer k such that n=k²”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and P, exactly the form of modus ponens, so Q follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-ponens-0009/)

Mathematical proof · qf-proof-modus-ponens · candidate





## 10. Modus Ponens

Let P mean “the rational number r equals 0” and Q mean “the absolute value of r equals 0”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and P, exactly the form of modus ponens, so Q follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-ponens-0010/)

Mathematical proof · qf-proof-modus-ponens · candidate





## 11. Modus Ponens

Let P mean “set A is empty” and Q mean “A has no elements”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and P, exactly the form of modus ponens, so Q follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-ponens-0011/)

Mathematical proof · qf-proof-modus-ponens · candidate





## 12. Modus Ponens

Let P mean “function f is constant” and Q mean “any two inputs have the same output”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and P, exactly the form of modus ponens, so Q follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-ponens-0012/)

Mathematical proof · qf-proof-modus-ponens · candidate





## 13. Modus Ponens

Let P mean “the angle measure is 90°” and Q mean “the angle is a right angle”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and P, exactly the form of modus ponens, so Q follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-ponens-0013/)

Mathematical proof · qf-proof-modus-ponens · candidate





## 14. Modus Ponens

Let P mean “the angle measure is 180°” and Q mean “the angle is a straight angle”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and P, exactly the form of modus ponens, so Q follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-ponens-0014/)

Mathematical proof · qf-proof-modus-ponens · candidate





## 15. Modus Ponens

Let P mean “the integer n is divisible by 8” and Q mean “the integer n is divisible by 2”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and P, exactly the form of modus ponens, so Q follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-ponens-0015/)

Mathematical proof · qf-proof-modus-ponens · candidate





## 16. Modus Ponens

Let P mean “the integer n is a multiple of 15” and Q mean “the integer n is divisible by 5”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and P, exactly the form of modus ponens, so Q follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-ponens-0016/)

Mathematical proof · qf-proof-modus-ponens · candidate





## 17. Modus Ponens

Let P mean “the number x is greater than 7” and Q mean “x is greater than 0”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and P, exactly the form of modus ponens, so Q follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-ponens-0017/)

Mathematical proof · qf-proof-modus-ponens · candidate





## 18. Modus Ponens

Let P mean “the number x is less than -3” and Q mean “x is less than 0”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and P, exactly the form of modus ponens, so Q follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-ponens-0018/)

Mathematical proof · qf-proof-modus-ponens · candidate





## 19. Modus Ponens

Let P mean “the polygon is a pentagon” and Q mean “the polygon has five sides”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and P, exactly the form of modus ponens, so Q follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-ponens-0019/)

Mathematical proof · qf-proof-modus-ponens · candidate





## 20. Modus Ponens

Let P mean “the rectangle has equal length and width” and Q mean “the rectangle is a square”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and P, exactly the form of modus ponens, so Q follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-ponens-0020/)

Mathematical proof · qf-proof-modus-ponens · candidate





## 21. Modus Ponens

Let P mean “the integer n is even” and Q mean “n² is even”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and P, exactly the form of modus ponens, so Q follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-ponens-0021/)

Mathematical proof · qf-proof-modus-ponens · candidate





## 22. Modus Ponens

Let P mean “the integer n is divisible by 3” and Q mean “n² is divisible by 3”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and P, exactly the form of modus ponens, so Q follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-ponens-0022/)

Mathematical proof · qf-proof-modus-ponens · candidate





## 23. Modus Ponens

Let P mean “the denominator b of a/b is nonzero” and Q mean “a/b is a defined fractional representation”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and P, exactly the form of modus ponens, so Q follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-ponens-0023/)

Mathematical proof · qf-proof-modus-ponens · candidate





## 24. Modus Ponens

Let P mean “set A is a subset of B” and Q mean “every element of A is also in B”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and P, exactly the form of modus ponens, so Q follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-ponens-0024/)

Mathematical proof · qf-proof-modus-ponens · candidate





## 25. Modus Ponens

Let P mean “propositions R and S both hold” and Q mean “proposition R holds”. The stated premises are P→Q and P, and the proposed conclusion is Q. Under modus ponens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and P, exactly the form of modus ponens, so Q follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-ponens-0025/)

Mathematical proof · qf-proof-modus-ponens · candidate





## 26. Modus Tollens

Let P mean “the integer n is divisible by 4” and Q mean “the integer n is even”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-tollens-0001/)

Mathematical proof · qf-proof-modus-tollens · candidate





## 27. Modus Tollens

Let P mean “the integer n is divisible by 6” and Q mean “the integer n is divisible by 3”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-tollens-0002/)

Mathematical proof · qf-proof-modus-tollens · candidate





## 28. Modus Tollens

Let P mean “the integer n is a multiple of 9” and Q mean “the integer n is a multiple of 3”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-tollens-0003/)

Mathematical proof · qf-proof-modus-tollens · candidate





## 29. Modus Tollens

Let P mean “the integer n is prime and greater than 2” and Q mean “the integer n is odd”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-tollens-0004/)

Mathematical proof · qf-proof-modus-tollens · candidate





## 30. Modus Tollens

Let P mean “the triangle has three equal sides” and Q mean “the triangle has at least two equal sides”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-tollens-0005/)

Mathematical proof · qf-proof-modus-tollens · candidate





## 31. Modus Tollens

Let P mean “the quadrilateral is a square” and Q mean “the quadrilateral has four right angles”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-tollens-0006/)

Mathematical proof · qf-proof-modus-tollens · candidate





## 32. Modus Tollens

Let P mean “the integer n is a multiple of 10” and Q mean “the integer n is a multiple of 5”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-tollens-0007/)

Mathematical proof · qf-proof-modus-tollens · candidate





## 33. Modus Tollens

Let P mean “the integer n is a multiple of 12” and Q mean “the integer n is a multiple of 4”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-tollens-0008/)

Mathematical proof · qf-proof-modus-tollens · candidate





## 34. Modus Tollens

Let P mean “the integer n is a perfect square” and Q mean “there exists an integer k such that n=k²”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-tollens-0009/)

Mathematical proof · qf-proof-modus-tollens · candidate





## 35. Modus Tollens

Let P mean “the rational number r equals 0” and Q mean “the absolute value of r equals 0”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-tollens-0010/)

Mathematical proof · qf-proof-modus-tollens · candidate





## 36. Modus Tollens

Let P mean “set A is empty” and Q mean “A has no elements”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-tollens-0011/)

Mathematical proof · qf-proof-modus-tollens · candidate





## 37. Modus Tollens

Let P mean “function f is constant” and Q mean “any two inputs have the same output”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-tollens-0012/)

Mathematical proof · qf-proof-modus-tollens · candidate





## 38. Modus Tollens

Let P mean “the angle measure is 90°” and Q mean “the angle is a right angle”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-tollens-0013/)

Mathematical proof · qf-proof-modus-tollens · candidate





## 39. Modus Tollens

Let P mean “the angle measure is 180°” and Q mean “the angle is a straight angle”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-tollens-0014/)

Mathematical proof · qf-proof-modus-tollens · candidate





## 40. Modus Tollens

Let P mean “the integer n is divisible by 8” and Q mean “the integer n is divisible by 2”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-tollens-0015/)

Mathematical proof · qf-proof-modus-tollens · candidate





## 41. Modus Tollens

Let P mean “the integer n is a multiple of 15” and Q mean “the integer n is divisible by 5”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-tollens-0016/)

Mathematical proof · qf-proof-modus-tollens · candidate





## 42. Modus Tollens

Let P mean “the number x is greater than 7” and Q mean “x is greater than 0”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-tollens-0017/)

Mathematical proof · qf-proof-modus-tollens · candidate





## 43. Modus Tollens

Let P mean “the number x is less than -3” and Q mean “x is less than 0”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-tollens-0018/)

Mathematical proof · qf-proof-modus-tollens · candidate





## 44. Modus Tollens

Let P mean “the polygon is a pentagon” and Q mean “the polygon has five sides”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-tollens-0019/)

Mathematical proof · qf-proof-modus-tollens · candidate





## 45. Modus Tollens

Let P mean “the rectangle has equal length and width” and Q mean “the rectangle is a square”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-tollens-0020/)

Mathematical proof · qf-proof-modus-tollens · candidate





## 46. Modus Tollens

Let P mean “the integer n is even” and Q mean “n² is even”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-tollens-0021/)

Mathematical proof · qf-proof-modus-tollens · candidate





## 47. Modus Tollens

Let P mean “the integer n is divisible by 3” and Q mean “n² is divisible by 3”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-tollens-0022/)

Mathematical proof · qf-proof-modus-tollens · candidate





## 48. Modus Tollens

Let P mean “the denominator b of a/b is nonzero” and Q mean “a/b is a defined fractional representation”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-tollens-0023/)

Mathematical proof · qf-proof-modus-tollens · candidate





## 49. Modus Tollens

Let P mean “set A is a subset of B” and Q mean “every element of A is also in B”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-tollens-0024/)

Mathematical proof · qf-proof-modus-tollens · candidate





## 50. Modus Tollens

Let P mean “propositions R and S both hold” and Q mean “proposition R holds”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?

View answer and explanationCan

The premises are P→Q and ¬Q, exactly the form of modus tollens, so ¬P follows.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-modus-tollens-0025/)

Mathematical proof · qf-proof-modus-tollens · candidate





## 51. Invalid Affirmation of the Consequent

Let P mean “the integer n is divisible by 4” and Q mean “the integer n is even”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?

View answer and explanationCannot

Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-invalid-converse-0001/)

Mathematical proof · qf-proof-invalid-converse · candidate





## 52. Invalid Affirmation of the Consequent

Let P mean “the integer n is divisible by 6” and Q mean “the integer n is divisible by 3”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?

View answer and explanationCannot

Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-invalid-converse-0002/)

Mathematical proof · qf-proof-invalid-converse · candidate





## 53. Invalid Affirmation of the Consequent

Let P mean “the integer n is a multiple of 9” and Q mean “the integer n is a multiple of 3”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?

View answer and explanationCannot

Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-invalid-converse-0003/)

Mathematical proof · qf-proof-invalid-converse · candidate





## 54. Invalid Affirmation of the Consequent

Let P mean “the integer n is prime and greater than 2” and Q mean “the integer n is odd”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?

View answer and explanationCannot

Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-invalid-converse-0004/)

Mathematical proof · qf-proof-invalid-converse · candidate





## 55. Invalid Affirmation of the Consequent

Let P mean “the triangle has three equal sides” and Q mean “the triangle has at least two equal sides”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?

View answer and explanationCannot

Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-invalid-converse-0005/)

Mathematical proof · qf-proof-invalid-converse · candidate





## 56. Invalid Affirmation of the Consequent

Let P mean “the quadrilateral is a square” and Q mean “the quadrilateral has four right angles”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?

View answer and explanationCannot

Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-invalid-converse-0006/)

Mathematical proof · qf-proof-invalid-converse · candidate





## 57. Invalid Affirmation of the Consequent

Let P mean “the integer n is a multiple of 10” and Q mean “the integer n is a multiple of 5”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?

View answer and explanationCannot

Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-invalid-converse-0007/)

Mathematical proof · qf-proof-invalid-converse · candidate





## 58. Invalid Affirmation of the Consequent

Let P mean “the integer n is a multiple of 12” and Q mean “the integer n is a multiple of 4”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?

View answer and explanationCannot

Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-invalid-converse-0008/)

Mathematical proof · qf-proof-invalid-converse · candidate





## 59. Invalid Affirmation of the Consequent

Let P mean “the integer n is a perfect square” and Q mean “there exists an integer k such that n=k²”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?

View answer and explanationCannot

Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-invalid-converse-0009/)

Mathematical proof · qf-proof-invalid-converse · candidate





## 60. Invalid Affirmation of the Consequent

Let P mean “the rational number r equals 0” and Q mean “the absolute value of r equals 0”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?

View answer and explanationCannot

Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-invalid-converse-0010/)

Mathematical proof · qf-proof-invalid-converse · candidate





## 61. Invalid Affirmation of the Consequent

Let P mean “set A is empty” and Q mean “A has no elements”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?

View answer and explanationCannot

Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-invalid-converse-0011/)

Mathematical proof · qf-proof-invalid-converse · candidate





## 62. Invalid Affirmation of the Consequent

Let P mean “function f is constant” and Q mean “any two inputs have the same output”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?

View answer and explanationCannot

Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-invalid-converse-0012/)

Mathematical proof · qf-proof-invalid-converse · candidate





## 63. Invalid Affirmation of the Consequent

Let P mean “the angle measure is 90°” and Q mean “the angle is a right angle”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?

View answer and explanationCannot

Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-invalid-converse-0013/)

Mathematical proof · qf-proof-invalid-converse · candidate





## 64. Invalid Affirmation of the Consequent

Let P mean “the angle measure is 180°” and Q mean “the angle is a straight angle”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?

View answer and explanationCannot

Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-invalid-converse-0014/)

Mathematical proof · qf-proof-invalid-converse · candidate





## 65. Invalid Affirmation of the Consequent

Let P mean “the integer n is divisible by 8” and Q mean “the integer n is divisible by 2”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?

View answer and explanationCannot

Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-invalid-converse-0015/)

Mathematical proof · qf-proof-invalid-converse · candidate





## 66. Invalid Affirmation of the Consequent

Let P mean “the integer n is a multiple of 15” and Q mean “the integer n is divisible by 5”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?

View answer and explanationCannot

Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-invalid-converse-0016/)

Mathematical proof · qf-proof-invalid-converse · candidate





## 67. Invalid Affirmation of the Consequent

Let P mean “the number x is greater than 7” and Q mean “x is greater than 0”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?

View answer and explanationCannot

Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-invalid-converse-0017/)

Mathematical proof · qf-proof-invalid-converse · candidate





## 68. Invalid Affirmation of the Consequent

Let P mean “the number x is less than -3” and Q mean “x is less than 0”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?

View answer and explanationCannot

Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-invalid-converse-0018/)

Mathematical proof · qf-proof-invalid-converse · candidate





## 69. Invalid Affirmation of the Consequent

Let P mean “the polygon is a pentagon” and Q mean “the polygon has five sides”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?

View answer and explanationCannot

Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-invalid-converse-0019/)

Mathematical proof · qf-proof-invalid-converse · candidate





## 70. Invalid Affirmation of the Consequent

Let P mean “the rectangle has equal length and width” and Q mean “the rectangle is a square”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?

View answer and explanationCannot

Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-invalid-converse-0020/)

Mathematical proof · qf-proof-invalid-converse · candidate





## 71. Invalid Affirmation of the Consequent

Let P mean “the integer n is even” and Q mean “n² is even”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?

View answer and explanationCannot

Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-invalid-converse-0021/)

Mathematical proof · qf-proof-invalid-converse · candidate





## 72. Invalid Affirmation of the Consequent

Let P mean “the integer n is divisible by 3” and Q mean “n² is divisible by 3”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?

View answer and explanationCannot

Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-invalid-converse-0022/)

Mathematical proof · qf-proof-invalid-converse · candidate





## 73. Invalid Affirmation of the Consequent

Let P mean “the denominator b of a/b is nonzero” and Q mean “a/b is a defined fractional representation”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?

View answer and explanationCannot

Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-invalid-converse-0023/)

Mathematical proof · qf-proof-invalid-converse · candidate





## 74. Invalid Affirmation of the Consequent

Let P mean “set A is a subset of B” and Q mean “every element of A is also in B”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?

View answer and explanationCannot

Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-invalid-converse-0024/)

Mathematical proof · qf-proof-invalid-converse · candidate





## 75. Invalid Affirmation of the Consequent

Let P mean “propositions R and S both hold” and Q mean “proposition R holds”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?

View answer and explanationCannot

Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-invalid-converse-0025/)

Mathematical proof · qf-proof-invalid-converse · candidate





## 76. Counterexample to a Universal Proposition

The domain is explicitly the integers. Consider the universal proposition “Every integer is even.” and the stated witness n=1. Does this witness refute that universal proposition?

View answer and explanationYes

n=1 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-counterexample-universal-0001/)

Mathematical proof · qf-proof-counterexample-universal · candidate





## 77. Counterexample to a Universal Proposition

The domain is explicitly the integers. Consider the universal proposition “Every integer is greater than 0.” and the stated witness n=0. Does this witness refute that universal proposition?

View answer and explanationYes

n=0 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-counterexample-universal-0002/)

Mathematical proof · qf-proof-counterexample-universal · candidate





## 78. Counterexample to a Universal Proposition

The domain is explicitly the integers. Consider the universal proposition “Every integer is greater than or equal to 0.” and the stated witness n=-1. Does this witness refute that universal proposition?

View answer and explanationYes

n=-1 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-counterexample-universal-0003/)

Mathematical proof · qf-proof-counterexample-universal · candidate





## 79. Counterexample to a Universal Proposition

The domain is explicitly the integers. Consider the universal proposition “Every integer is a multiple of 3.” and the stated witness n=1. Does this witness refute that universal proposition?

View answer and explanationYes

n=1 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-counterexample-universal-0004/)

Mathematical proof · qf-proof-counterexample-universal · candidate





## 80. Counterexample to a Universal Proposition

The domain is explicitly the integers. Consider the universal proposition “Every integer is a multiple of 5.” and the stated witness n=2. Does this witness refute that universal proposition?

View answer and explanationYes

n=2 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-counterexample-universal-0005/)

Mathematical proof · qf-proof-counterexample-universal · candidate





## 81. Counterexample to a Universal Proposition

The domain is explicitly the integers. Consider the universal proposition “Every integer equals 0.” and the stated witness n=1. Does this witness refute that universal proposition?

View answer and explanationYes

n=1 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-counterexample-universal-0006/)

Mathematical proof · qf-proof-counterexample-universal · candidate





## 82. Counterexample to a Universal Proposition

The domain is explicitly the integers. Consider the universal proposition “Every integer is greater than 5.” and the stated witness n=5. Does this witness refute that universal proposition?

View answer and explanationYes

n=5 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-counterexample-universal-0007/)

Mathematical proof · qf-proof-counterexample-universal · candidate





## 83. Counterexample to a Universal Proposition

The domain is explicitly the integers. Consider the universal proposition “Every integer is less than 10.” and the stated witness n=10. Does this witness refute that universal proposition?

View answer and explanationYes

n=10 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-counterexample-universal-0008/)

Mathematical proof · qf-proof-counterexample-universal · candidate





## 84. Counterexample to a Universal Proposition

The domain is explicitly the integers. Consider the universal proposition “Every integer is even.” and the stated witness n=-3. Does this witness refute that universal proposition?

View answer and explanationYes

n=-3 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-counterexample-universal-0009/)

Mathematical proof · qf-proof-counterexample-universal · candidate





## 85. Counterexample to a Universal Proposition

The domain is explicitly the integers. Consider the universal proposition “Every integer is greater than 0.” and the stated witness n=-1. Does this witness refute that universal proposition?

View answer and explanationYes

n=-1 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-counterexample-universal-0010/)

Mathematical proof · qf-proof-counterexample-universal · candidate





## 86. Counterexample to a Universal Proposition

The domain is explicitly the integers. Consider the universal proposition “Every integer is a multiple of 7.” and the stated witness n=3. Does this witness refute that universal proposition?

View answer and explanationYes

n=3 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-counterexample-universal-0011/)

Mathematical proof · qf-proof-counterexample-universal · candidate





## 87. Counterexample to a Universal Proposition

The domain is explicitly the integers. Consider the universal proposition “Every integer equals 0.” and the stated witness n=-2. Does this witness refute that universal proposition?

View answer and explanationYes

n=-2 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-counterexample-universal-0012/)

Mathematical proof · qf-proof-counterexample-universal · candidate





## 88. Counterexample to a Universal Proposition

The domain is explicitly the integers. Consider the universal proposition “Every integer is greater than -4.” and the stated witness n=-4. Does this witness refute that universal proposition?

View answer and explanationYes

n=-4 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-counterexample-universal-0013/)

Mathematical proof · qf-proof-counterexample-universal · candidate





## 89. Counterexample to a Universal Proposition

The domain is explicitly the integers. Consider the universal proposition “Every integer is less than 0.” and the stated witness n=0. Does this witness refute that universal proposition?

View answer and explanationYes

n=0 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-counterexample-universal-0014/)

Mathematical proof · qf-proof-counterexample-universal · candidate





## 90. Counterexample to a Universal Proposition

The domain is explicitly the integers. Consider the universal proposition “Every integer is a multiple of 2.” and the stated witness n=5. Does this witness refute that universal proposition?

View answer and explanationYes

n=5 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-counterexample-universal-0015/)

Mathematical proof · qf-proof-counterexample-universal · candidate





## 91. Counterexample to a Universal Proposition

The domain is explicitly the integers. Consider the universal proposition “Every integer is greater than or equal to 0.” and the stated witness n=-9. Does this witness refute that universal proposition?

View answer and explanationYes

n=-9 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-counterexample-universal-0016/)

Mathematical proof · qf-proof-counterexample-universal · candidate





## 92. Counterexample to a Universal Proposition

The domain is explicitly the integers. Consider the universal proposition “Every integer is a multiple of 11.” and the stated witness n=4. Does this witness refute that universal proposition?

View answer and explanationYes

n=4 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-counterexample-universal-0017/)

Mathematical proof · qf-proof-counterexample-universal · candidate





## 93. Counterexample to a Universal Proposition

The domain is explicitly the integers. Consider the universal proposition “Every integer is even.” and the stated witness n=9. Does this witness refute that universal proposition?

View answer and explanationYes

n=9 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-counterexample-universal-0018/)

Mathematical proof · qf-proof-counterexample-universal · candidate





## 94. Counterexample to a Universal Proposition

The domain is explicitly the integers. Consider the universal proposition “Every integer is greater than 0.” and the stated witness n=-8. Does this witness refute that universal proposition?

View answer and explanationYes

n=-8 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-counterexample-universal-0019/)

Mathematical proof · qf-proof-counterexample-universal · candidate





## 95. Counterexample to a Universal Proposition

The domain is explicitly the integers. Consider the universal proposition “Every integer is greater than 100.” and the stated witness n=100. Does this witness refute that universal proposition?

View answer and explanationYes

n=100 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-counterexample-universal-0020/)

Mathematical proof · qf-proof-counterexample-universal · candidate





## 96. Counterexample to a Universal Proposition

The domain is explicitly the integers. Consider the universal proposition “Every integer is less than -2.” and the stated witness n=-2. Does this witness refute that universal proposition?

View answer and explanationYes

n=-2 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-counterexample-universal-0021/)

Mathematical proof · qf-proof-counterexample-universal · candidate





## 97. Counterexample to a Universal Proposition

The domain is explicitly the integers. Consider the universal proposition “Every integer is a multiple of 4.” and the stated witness n=6. Does this witness refute that universal proposition?

View answer and explanationYes

n=6 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-counterexample-universal-0022/)

Mathematical proof · qf-proof-counterexample-universal · candidate





## 98. Counterexample to a Universal Proposition

The domain is explicitly the integers. Consider the universal proposition “Every integer equals 0.” and the stated witness n=12. Does this witness refute that universal proposition?

View answer and explanationYes

n=12 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-counterexample-universal-0023/)

Mathematical proof · qf-proof-counterexample-universal · candidate





## 99. Counterexample to a Universal Proposition

The domain is explicitly the integers. Consider the universal proposition “Every integer is greater than or equal to 0.” and the stated witness n=-12. Does this witness refute that universal proposition?

View answer and explanationYes

n=-12 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-counterexample-universal-0024/)

Mathematical proof · qf-proof-counterexample-universal · candidate





## 100. Counterexample to a Universal Proposition

The domain is explicitly the integers. Consider the universal proposition “Every integer is a multiple of 9.” and the stated witness n=10. Does this witness refute that universal proposition?

View answer and explanationYes

n=10 lies in the integer domain and fails the required predicate, so it is a valid counterexample to this stated universal proposition only.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260921-proof-counterexample-universal-0025/)

Mathematical proof · qf-proof-counterexample-universal · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20260921-proof-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20260921-proof-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20260921-proof-foundations/validation-report.json)