Contraposition of "P if and only if Q"

Question

I'm understanding the basic idea of contraposition, when it comes to propositional logic and writing proofs, but I'm having trouble figuring out what the contraposition of "P if and only if Q" would be. It seems simple enough that the contraposition of "If P then Q" would be "If not Q then not P", but that seems too simple to be the case with if and only if. Anyone able to help?

Answer 1

$P \Leftrightarrow Q$ is the same as $P \Rightarrow Q$ and $Q \Rightarrow P$. So contrapose those two and you get $Not(Q) \Rightarrow Not(P)$ and $Not(P) \Rightarrow Not(Q)$, otherwise written as $Not(P) \Leftrightarrow Not(Q)$

Answer 2

Contrapositives are typically understood to be for conditionals. That is, conjunctions and disjunctions don't have any contrapositive. and the biconditinal ... well ... that's unusual too.

BUT it *is * true that $P \leftrightarrow Q \Leftrightarrow \neg Q \leftrightarrow \neg P$, so you could see this as the Contrapositive of the biconditional ... But again, that's pretty unusal. There will probably be a few websites, and maybe even some textbooks that talk about the contrapositive of a biconditional this way, but it is not standard nomenclature.