# Converse, Inverse, and Contrapositive of Conditional Statement

What we want to achieve in this lesson is to be familiar with the fundamental rules on how to convert or rewrite a conditional statement into its converse, inverse, and contrapositive.

But first, we need to review what a conditional statement is because it is the foundation or precursor of the three related sentences that we are going to discuss in this lesson.

Given an if-then statement “if pp , then qq ,” we can create three related statements:

A conditional statement consists of two parts, a hypothesis in the “if” clause and a conclusion in the “then” clause.  For instance, “If it rains, then they cancel school.”
“It rains” is the hypothesis.
“They cancel school” is the conclusion.

To form the converse of the conditional statement, interchange the hypothesis and the conclusion.
The converse of “If it rains, then they cancel school” is “If they cancel school, then it rains.”

To form the inverse of the conditional statement, take the negation of both the hypothesis and the conclusion.
The inverse of “If it rains, then they cancel school” is “If it does not rain, then they do not cancel school.”

To form the contrapositive of the conditional statement, interchange the hypothesis and the conclusion of the inverse statement.
The contrapositive of “If it rains, then they cancel school” is “If they do not cancel school, then it does not rain.”

If the statement is true, then the contrapositive is also logically true. If the converse is true, then the inverse is also logically true.

Example 1:

In the above example, since the hypothesis and conclusion are equivalent, all four statements are true. But this will not always be the case!

Example 2: