Converse of Conditional is Contrapositive of Inverse
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Theorem
Let $p \implies q$ be a conditional.
Then the converse of $p \implies q$ is the contrapositive of its inverse.
Proof
The converse of $p \implies q$ is:
- $q \implies p$
The inverse of $p \implies q$ is:
- $\neg p \implies \neg q$
The contrapositive of $\neg p \implies \neg q$ is:
- $\neg \neg q \implies \neg \neg p$
By Double Negation, the two are seen to be equal.
$\blacksquare$