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

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!

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!