The Contrapositive, Converse, and Inverse of an Implication
The Contrapositive, Converse, and Inverse of an Implication
| Definition: Let $P$ and $Q$ be statements and consider the implication $P \rightarrow Q$. The Contrapositive of this implication is the formula $\neg Q \rightarrow \neg P$. The Converse of this implication is the formula $Q \rightarrow P$. The Inverse of this implication is the formula $\neg P \rightarrow \neg Q$. |
The implication $P \rightarrow Q$ and the contrapositive $\neg Q \rightarrow \neg P$ have the property that they are logically equivalent which we prove below.
| Proposition 1: Let $P$ and $Q$ be statements. Then $(P \rightarrow Q) \Leftrightarrow (\neg Q \rightarrow \neg P)$, that is, the implication $P \rightarrow Q$ is logically equivalent to the contrapositive $\neg Q \rightarrow \neg P$. |
- Proof: We construct the truth tables $P \rightarrow Q$ and $\neg Q \rightarrow \neg P$:
| $P$ | $Q$ | $P \rightarrow Q$ |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
| $P$ | $Q$ | $\neg P$ | $\neg Q$ | $\neg Q \rightarrow \neg P$ |
|---|---|---|---|---|
| T | T | F | F | T |
| T | F | F | T | F |
| F | T | T | F | T |
| F | F | T | T | T |
- Comparing the far right columns of the truth tables above and we conclude that $(P \rightarrow Q) \Leftrightarrow (\neg Q \rightarrow \neg P)$. $\blacksquare$
Similarly, the converse and inverse of the implication $P \rightarrow Q$ are logically equivalent.
| Proposition 2: Let $P$ and $Q$ be statements. Then $(Q \rightarrow P) \Leftrightarrow (\neg P \rightarrow \neg Q)$, that is, the converse $Q \rightarrow P$ is logically equivalent to the inverse $\neg P \rightarrow \neg Q$ |
- Proof: This follows immediately by proposition 1 by a change of variables. $\blacksquare$
Let $P$ be the statement "Bob is hungry" and let $Q$ be the statement "Bob will eat lunch". The implication $P \rightarrow Q$, its contrapositive, converse, and inverse are all listed below:
- $P \rightarrow Q$: If Bob is hungry then Bob will eat lunch.
- $\neg Q \rightarrow \neg P$: If Bob doesn't eat lunch then Bob is not hungry.
- $Q \rightarrow P$: If Bob eats lunch then Bob is hungry.
- $\neg P \rightarrow \neg Q$: If Bob is not hungry then Bob will not eat lunch.