# Implication

'If P then Q'. It is false only when P is true and Q is false. In every other case it counts as true.

Maths (university) · Part 01: Logic, sets and proof · Card 2 of 9 · Last updated 2026-10-19

## Analogy

A promise: 'if it rains, I will bring an umbrella'. You only break it if it rains and you come without one.

## Example

**If x > 3 then x^2 > 9**

x = 4: 16 > 9, so it holds.
x = 2: P is false, so no test.
A counterexample needs P true
and Q false at once.

## In practice

Most theorems have the form 'if assumptions then conclusion', so you must read implications exactly.

## Practice

### Flashcards

- **When is 'if P then Q' false?** Only when P is true and Q is false.
- **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.
- **What does one counterexample show?** It disproves a 'for all' claim completely.

## Questions people ask

### Why is 'if P then Q' true when P is false?

Because the only way to break the promise is P true with Q false. If P never happens, nothing was broken. This is called vacuous truth. For example, 'if 2 is odd then 5 = 7' is a true implication, since the condition fails from the start.

### How do you disprove an implication?

Find one counterexample: a case where P is true and Q is false. To disprove 'if n is prime then n is odd', take n = 2. It is prime but even. One counterexample is enough, while examples that fit never prove a general claim.

## Related concepts

- [Contrapositive](https://studyvisually.com/learn/science/maths-university/maths-university-part-1/maths-university-part-1-concept-3)
- [Quantifiers](https://studyvisually.com/learn/science/maths-university/maths-university-part-1/maths-university-part-1-concept-4)
- [Statement](https://studyvisually.com/learn/science/maths-university/maths-university-part-1/maths-university-part-1-concept-1)

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