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?
Contraposition of "P if and only if Q"
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logic
proof-writing
propositional-calculus
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0You have that $P$ implies $Q$ and you have found the correct contraposition of this. You also have $Q$ implies $P$. what is the contraposition of this last statement? Throw al this information together to find the contraposition of an equivalente (if and only if statement). – 2017-02-07
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0not Q if and only if not P. – 2017-02-07
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1The term "contrapositive" really only applies to "if/then" statements, not to "if and only if" statements. – 2017-02-07
2 Answers
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$P \Leftrightarrow Q$ 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 $Not(P) \Leftrightarrow Not(Q)$
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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.