Contrapositive Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Contrapositive.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The statement 'If not Q, then not P'—logically equivalent to 'If P, then Q.'

Flip and negate. Always has the same truth value as the original.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The contrapositive \neg Q \to \neg P is logically equivalent to P \to Q — they always have the same truth value. Proving one proves the other.

Common stuck point: Contrapositive \neq converse. Converse: 'If Q, then P'—NOT equivalent.

Sense of Study hint: Write the original as 'If P then Q.' Now swap P and Q to get the converse, then negate both to get the contrapositive. Keep those two straight.

Worked Examples

Example 1

easy
Form the contrapositive of: 'If n^2 is even, then n is even.'

Solution

  1. 1
    Recall the structure of the conditional: p \Rightarrow q where p: 'n^2 is even' and q: 'n is even.'
  2. 2
    The contrapositive is \neg q \Rightarrow \neg p. Form the negations: \neg q: 'n is not even' (i.e., n is odd), and \neg p: 'n^2 is not even' (i.e., n^2 is odd).
  3. 3
    Contrapositive: 'If n is odd, then n^2 is odd.' By the logical equivalence p \Rightarrow q \equiv \neg q \Rightarrow \neg p, this statement has exactly the same truth value as the original — and it is in fact true (odd times odd is odd).

Answer

\text{If } n \text{ is odd, then } n^2 \text{ is odd.}
The contrapositive negates both parts and swaps them. It is always logically equivalent to the original conditional, making it a powerful tool in proofs.

Example 2

medium
Prove by contrapositive: 'If n^2 is odd, then n is odd.'

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Write the contrapositive of: 'If a triangle is equilateral, then all its angles are 60°.'

Example 2

medium
Write the contrapositive of: 'If n^2 is even, then n is even.'
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Background Knowledge

These ideas may be useful before you work through the harder examples.

conditionalnegation