# Contrapositive

The implication 'if not Q then not P'. It is true exactly when 'if P then Q' is true, so you can prove either one.

Maths (university) · Part 01: Logic, sets and proof · Card 3 of 9 · Last updated 2026-10-19

## Analogy

Reading a map backwards: if every road from A leads to B, then anyone not at B cannot have come from A.

## Example

**P: 4 divides n. Q: n is even.**

Original: 4 | n gives n even.
Contrapositive: n odd gives 4 does not divide n.
Converse: n even gives 4 | n.
The converse fails for n = 2.

## In practice

Use it when the 'not' version is easier to start from. Do not confuse it with the converse, which is a different claim.

## Practice

### Flashcards

- **What is the contrapositive of 'if P then Q'?** 'If not Q then not P'. It always has the same truth value as the original.
- **Is the converse of a true implication always true?** No. 'If 4 divides n then n is even' is true, but its converse is false for n = 2.

## Questions people ask

### What is the difference between the contrapositive and the converse?

The contrapositive of 'if P then Q' is 'if not Q then not P' and always has the same truth value as the original. The converse is 'if Q then P' and can differ. 'If n is divisible by 4 then n is even' is true, but its converse is false.

### When should I prove the contrapositive instead of the original?

Use it when the conclusion's negation gives you something concrete to work with. To show 'if n squared is even then n is even', assume n is odd, so n = 2k + 1. Then n squared is odd, and the original claim follows.

## Related concepts

- [Implication](https://studyvisually.com/learn/science/maths-university/maths-university-part-1/maths-university-part-1-concept-2)
- [Statement](https://studyvisually.com/learn/science/maths-university/maths-university-part-1/maths-university-part-1-concept-1)
- [Quantifiers](https://studyvisually.com/learn/science/maths-university/maths-university-part-1/maths-university-part-1-concept-4)
- [Direct proof](https://studyvisually.com/learn/science/maths-university/maths-university-part-1/maths-university-part-1-concept-5)

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