Contrapositive Statement

How to find the contrapositive of a conditional statement: definition, 2 examples, and their solutions.

Definition

Definition

[~q → ~p] is the contrapositive (contraposition)
of the conditional statement [p → q].

To find the contrapositive,
switch and negate both p and q.

Example 1

Example

Solution

The structure of the given conditional is
[if ... , then ...].

So the statement behind if is p:
2 is a prime number.

And the statement behind then is q:
2 is an odd number.

The given conditional statement is p → q.

Then the contrapositive is,
switch and negate both p and q,
~q → ~p.

Write the contrapositive ~q → ~p.

If, not q, 2 is not an odd number,

then, not p, 2 is not a prime number.

So the contrapositive is
if 2 is not an odd number,
then 2 is not a prime number.

Example 2

Example

Solution

The structure of the given conditional is
[if ... , then ...].

So the statement behind if is p:
he is not in his room.

And the statement behind then is q:
he is playing basketball.

The given conditional statement is p → q.

Then the contrapositive is,
switch and negate both p and q,
~q → ~p.

Write the contrapositive ~q → ~p.

If, not q, he is not playing basketball,

then, not p, he is in his room.

So the contrapositive is
if he is not playing basketball,
then he is in his room.