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## Proof TechniqueCPContrapositives

The contrapositive of an implication $$P\Rightarrow Q$$ is the implication $${\rm not}(Q)\Rightarrow{\rm not}(P)\text{,}$$ where “not” means the logical negation, or opposite. An implication is true if and only if its contrapositive is true. In symbols, $$(P\Rightarrow Q)\iff({\rm not}(Q)\Rightarrow{\rm not}(P))$$ is a theorem. Such statements about logic, that are always true, are known as tautologies.

For example, it is a theorem that “if a vehicle is a fire truck, then it has big tires and has a siren.” (Yes, I'm sure you can conjure up a counterexample, but play along with me anyway.) The contrapositive is “if a vehicle does not have big tires or does not have a siren, then it is not a fire truck.” Notice how the “and” became an “or” when we negated the conclusion of the original theorem.

It will frequently happen that it is easier to construct a proof of the contrapositive than of the original implication. If you are having difficulty formulating a proof of some implication, see if the contrapositive is easier for you. The trick is to construct the negation of complicated statements accurately. More on that later.