Chapter 2 Direct Proofs
In order to prove that a conditional statement \(P \to Q\) is true, we only need to prove that \(Q\) is true whenever \(P\) is true. This is because the conditional statement is true whenever the hypothesis is false. So in a direct proof of \(P \to Q\text{,}\) we assume that \(P\) is true, and using this assumption, we proceed through a logical sequence of steps to arrive at the conclusion that \(Q\) is true. Unfortunately, it is often not easy to discover how to start this logical sequence of steps or how to get to the conclusion that \(Q\) is true. We will describe a method of exploration that often can help in discovering the steps of a proof. This method will involve working forward from the hypothesis, \(P\text{,}\) and backward from the conclusion, \(Q\text{.}\) We will illustrate this “forward-backward” method with the following proposition.