Practice Contrapositive in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

The contrapositive of a conditional statement P⇒QP \Rightarrow Q is ¬Q⇒¬P\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.

Showing a random 20 of 50 problems.

Example 1

medium
Write the contrapositive of 'If x∈A∩Bx \in A \cap B, then x∈Ax \in A.'

Example 2

easy
Write the converse of 'P⇒QP \Rightarrow Q' and state how it differs from the contrapositive.

Example 3

easy
Write the inverse of 'If x=0x = 0, then x2=0x^2 = 0.'

Example 4

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 5

hard
Build a truth table to verify that (P⇒Q)⇔(¬Q⇒¬P)(P \Rightarrow Q) \Leftrightarrow (\neg Q \Rightarrow \neg P).

Example 6

medium
Prove by contrapositive: 'If n2n^2 is odd, then nn is odd.'

Example 7

easy
Write the contrapositive of 'P⇒QP \Rightarrow Q'.

Example 8

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

Example 9

easy
Which form is the contrapositive of P⇒QP \Rightarrow Q?

Example 10

easy
Form the contrapositive of: 'If n2n^2 is even, then nn is even.'

Example 11

medium
Is the contrapositive of a FALSE conditional true or false?

Example 12

easy
Write the converse of 'If nn is prime, then n≥2n \ge 2.'

Example 13

challenge
Prove that P⇒QP \Rightarrow Q and its contrapositive ¬Q⇒¬P\neg Q \Rightarrow \neg P are logically equivalent using a truth table.

Example 14

medium
Given 'If x2−4>0x^2 - 4 > 0, then x>2x > 2 or x<−2x < -2', write the contrapositive.

Example 15

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 16

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

Example 17

challenge
Use the contrapositive to prove: if n2n^2 is even, then nn is even.

Example 18

medium
If 'If n2>25n^2 > 25 then n>5n > 5 or n<−5n < -5', form the contrapositive.

Example 19

easy
Is the contrapositive of a true conditional always true?

Example 20

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