2.3: Converse, Inverse, and Contrapositive

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Related to the conditional $$p \rightarrow q$$ are three important variations.

Definition: Converse

$$\displaystyle q \rightarrow p$$

Definition: Inverse

$$\displaystyle \neg p \rightarrow \neg q$$

Definition: Contrapositive

$$\displaystyle \neg q \rightarrow \neg p$$

Theorem $$\PageIndex{1}$$: Modus Tollens

A conditional and its contrapositive are equivalent.

Proof

We simply compare the truth tables.

 $$p$$ $$q$$ $$\neg p$$ $$\neg q$$ $$p\rightarrow q$$ $$\neg q \rightarrow \neg p$$ $$T$$ $$T$$ $$F$$ $$F$$ $$T$$ $$T$$ $$T$$ $$F$$ $$F$$ $$T$$ $$F$$ $$F$$ $$F$$ $$T$$ $$T$$ $$F$$ $$T$$ $$T$$ $$F$$ $$F$$ $$T$$ $$T$$ $$T$$ $$T$$

As the two “output” columns are identical, we conclude that the statements are equivalent.

Corollary $$\PageIndex{1}$$: Modus Tollens for Inverse and Converse

The inverse and converse of a conditional are equivalent.

Proof

The inverse of the conditional $$p \rightarrow q$$ is $$\neg p \rightarrow \neg q\text{.}$$ The contrapositive of this new conditional is $$\neg \neg q \rightarrow \neg \neg p\text{,}$$ which is equivalent to $$q \rightarrow p$$ by double negation.

Warning $$\PageIndex{1}$$: Common Mistakes

• Mixing up a conditional and its converse.
• Assuming that a conditional and its converse are equivalent.

Example $$\PageIndex{1}$$: Related Conditionals are not All Equivalent

1. Suppose $$m$$ is a fixed but unspecified whole number that is greater than $$2\text{.}$$

 conditional If $$m$$ is a prime number, then it is an odd number. contrapositive If $$m$$ is not an odd number, then it is not a prime number. converse If $$m$$ is an odd number, then it is a prime number. inverse If $$m$$ is not a prime number, then it is not an odd number.

Only two of these four statements are true!

1. Suppose $$f(x)$$ is a fixed but unspecified function.

 conditional If $$f$$ is continuous, then it is differentiable. contrapositive If $$f$$ is not differentiable, then it is not continuous. converse If $$f$$ is differentiable, then it is continuous. inverse If $$f$$ is not continuous, then it is not differentiable.

Only two of these four statements are true!

This page titled 2.3: Converse, Inverse, and Contrapositive is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Jeremy Sylvestre via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.