Question Video: Identifying the Converse, Inverse, and Contrapositive of a Conditional Statement Mathematics • 11th Grade

Read the following conditional statement: If it is raining, then Amelia has her umbrella up. Write the converse of the statement. Write the inverse of the statement. Write the contrapositive of the statement.

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Video Transcript

Read the following conditional statement: If it is raining, then Amelia has her umbrella up. Write the converse of the statement. Write the inverse of the statement. Write the contrapositive of the statement.

Here, we have our statement: “If it is raining, then Amelia has her umbrella up.” This is an if-then statement. If it is raining, then Amelia has her umbrella up. The converse statement will switch the if-then pieces. The converse statement would say if Amelia has her umbrella up, then it is raining.

The inverse is a little bit different. It’s also an if-then statement. But it’s the negated if-then statement. It’s an if not, then not. It would say if it is not raining, then Amelia does not have her umbrella up. And finally, the contrapositive is also if not, then not. But it switches the two pieces. The contrapositive would say if Amelia does not have her umbrella up, then it is not raining.

If-then statements are made up of two pieces, the hypothesis and the conclusion. Here, the hypothesis is it is raining. And the conclusion would be Amelia has her umbrella up. We’ll sometimes use the letter 𝑝 for the hypothesis and the letter 𝑞 for the conclusion.

A statement would be if 𝑝, then 𝑞. The converse is if 𝑞, then 𝑝. The inverse is if not 𝑝, then not 𝑞. And the contrapositive is if not 𝑞, then not 𝑝. So for the converse, we need to switch our conclusion and our hypothesis. We’ll say if Amelia has her umbrella up, then it’s raining.

The inverse will negate the hypothesis and negate the conclusion. If it is not raining, then Amelia does not have her umbrella up. And finally, the contrapositive: it negates the hypothesis and the conclusion and flips their order. If Amelia does not have her umbrella up, then it is not raining.