# Statement

A sentence that is definitely true or definitely false, never both. Maths proofs are built from statements.

Maths (university) · Part 01: Logic, sets and proof · Card 1 of 9 · Last updated 2026-10-19

## Analogy

A light switch: it is on or off, with no in-between.

## Example

**True or false?**

'7 is prime' is true.
'9 is prime' is false, since 9 = 3 x 3.
'x + 2 = 5' is not a statement yet.
It is true for x = 3, false for x = 4.

## In practice

Before you try to prove something, check it is a statement. 'x + 2 = 5' only becomes one once x is chosen.

## Questions people ask

### What is the difference between a statement and an open sentence?

A statement has a fixed truth value: true or false. An open sentence such as 'x is even' contains a free variable, so it has no truth value until you pick x or quantify it. Replacing x with 4 gives a true statement; with 5, a false one.

### Can a statement be both true and false in maths?

No. Classical mathematics assumes every statement is exactly one of true or false. This is why proof by contradiction works: if assuming something leads to both a statement and its opposite being true, the assumption cannot hold. Ambiguous everyday sentences are not allowed as maths statements, though.

## Related concepts

- [Implication](https://studyvisually.com/learn/science/maths-university/maths-university-part-1/maths-university-part-1-concept-2)
- [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)

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