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 contrapositive of a conditional statement $P \Rightarrow Q$ is $\neg Q \Rightarrow \neg P$ , formed by negating both parts and reversing their order — it is always logically equivalent to the original.

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 of if P then Q is if not Q then not P, always equivalent.

Common stuck point: The procedure for contrapositive is the easy part; the trap is negating without reversing, giving the inverse $\neg P \to \neg Q$ . Asking "Did I negate BOTH parts AND reverse their order, keeping the same truth value?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Did I negate BOTH parts AND reverse their order, keeping the same truth value?

Worked Examples

Example 1

easy

Form the contrapositive of: 'If

n^2

is even, then

n

is even.'

Answer

\text{If } n \text{ is odd, then } n^2 \text{ is odd.}

First step

Recall the structure of the conditional:

p \Rightarrow q

where

p

: '

n^2

is even' and

q

: '

n

is even.'

Full solution

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
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).

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.'

Example 3

medium

Prove by contrapositive: if

3n + 2

is odd, then

n

is odd.

Example 4

hard

Prove by contrapositive: if

n

is not divisible by 3, then

n^2

is not divisible by 3.

Example 5

hard

Build a truth table to verify that

(P \Rightarrow Q) \Leftrightarrow (\neg Q \Rightarrow \neg P)

Example 6

challenge

A student writes: 'If a triangle has two equal sides, then it has two equal angles' (true). They claim 'If a triangle does not have two equal sides, then it does not have two equal angles' is the contrapositive. Identify and correct the error.

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.'

Example 3

easy

Write the contrapositive of '

P \Rightarrow Q

Example 4

easy

Write the contrapositive of 'if it is a dog, then it is a mammal'.

Example 5

easy

Is the contrapositive of a true conditional always true?

Example 6

easy

Write the converse of '

P \Rightarrow Q

' and state how it differs from the contrapositive.

Example 7

easy

Write the inverse of '

P \Rightarrow Q

Example 8

easy

Write the contrapositive of 'if

x > 5

, then

x > 3

Example 9

easy

The contrapositive of '

\neg Q \Rightarrow \neg P

' is what?

Example 10

easy

Which is logically equivalent to '

P \Rightarrow Q

': its converse or its contrapositive?

Example 11

medium

Write the contrapositive of 'if a number is divisible by 6, then it is divisible by 3'.

Example 12

medium

Given the true statement 'if

n^2

is odd, then

n

is odd', write the contrapositive and confirm it is true.

Example 13

medium

State the contrapositive of 'if a shape is a square, then it has four equal sides', and the converse. Which is equivalent to the original?

Example 14

medium

The contrapositive of a statement is 'if

x

is not positive, then

x^2 \ne 4

'. What was the original statement?

Example 15

medium

Is the contrapositive of a FALSE conditional true or false?

Example 16

medium

Write the contrapositive of 'if a person lives in Paris, then they live in France'.

Example 17

medium

Match the inverse '

\neg P \Rightarrow \neg Q

' to the form it is equivalent to among: original, converse, contrapositive.

Example 18

challenge

Prove that

P \Rightarrow Q

and its contrapositive

\neg Q \Rightarrow \neg P

are logically equivalent using a truth table.

Example 19

challenge

Use the contrapositive to prove: if

n^2

is even, then

n

is even.

Example 20

challenge

A student claims 'if a number is not divisible by 6, then it is not divisible by 3' follows from 'if divisible by 6, then divisible by 3'. Identify the error.

Example 21

medium

Write the contrapositive of 'if today is Sunday, then the store is closed', then state what it lets you conclude if the store is open.

Example 22

medium

Write the contrapositive of 'if

a = 0

, then

ab = 0

Example 23

easy

Write the contrapositive of 'If it is raining, then the ground is wet.'

Example 24

easy

Write the contrapositive of 'If

x > 10

, then

x > 0

Example 25

easy

Write the converse of 'If

n

is prime, then

n \ge 2

Example 26

easy

Write the inverse of 'If

x = 0

, then

x^2 = 0

Example 27

medium

Write the contrapositive of 'If

f

is differentiable at

a

, then

f

is continuous at

a

Example 28

medium

Write the contrapositive of 'If a quadrilateral is a square, then it is a rectangle.'

Example 29

medium

Given 'If

x^2 - 4 > 0

, then

x > 2

x < -2

', write the contrapositive.

Example 30

medium

Write the contrapositive of 'If

ab = 0

, then

a = 0

b = 0

Example 31

medium

If 'If

n^2 > 25

then

n > 5

n < -5

', form the contrapositive.

Example 32

medium

Write the contrapositive of 'If a function is integrable on

[a,b]

, then it is bounded on

[a,b]

Example 33

medium

Write the contrapositive of 'If a polygon is regular, then all its sides are equal.'

Example 34

hard

The contrapositive of a statement is 'If a number does not end in 0 or 5, then it is not divisible by 5.' What is the original?

Example 35

hard

Use the contrapositive to prove: if

x + y

is irrational, then at least one of

x,y

is irrational.

Example 36

medium

Write the contrapositive of 'If

x \in A \cap B

, then

x \in A

Example 37

medium

Write the contrapositive of 'If

\gcd(a,b) = 1

, then

a

and

b

have no common prime factor.'

Example 38

challenge

Form the contrapositive of 'For all integers

n

, if

n^3

is even, then

n

is even', and prove it.

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Related Concepts

Conditional Statement Negation Proof Techniques

Background Knowledge

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

conditionalnegation