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We could go on and on and on about different proof styles (we haven't even mentioned induction or combinatorial proofs here), but instead we will end with one final useful technique: proof by cases. The idea is to prove that \(P\) is true by proving that \(Q \imp P\) and \(\neg Q \imp P\) for some statement \(Q\text{.}\) So no matter what, whether or not \(Q\) is true, we know that \(P\) is true. In fact, we could generalize this. Suppose we want to prove \(P\text{.}\) We know that at least one of the statements \(Q_1, Q_2, \ldots, Q_n\) is true. If we can show that \(Q_1 \imp P\) and \(Q_2 \imp P\) and so on all the way to \(Q_n \imp P\text{,}\) then we can conclude \(P\text{.}\) The key thing is that we want to be sure that one of our cases (the \(Q_i\)'s) must be true no matter what.