2.3: Converse, Inverse, and Contrapositive
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
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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!