Exercise 1.7

If we take the contrapositive of the statement

\[x^2 \text{ is odd} \Rightarrow x \text{ is odd}\]

then we should end up with

\[x \text{ is not odd} \Rightarrow x^2 \text{ is not odd}.\]

A more sensible way of saying “\(x\) is not odd” is to say “\(x\) is even”, so this statement is equivalent to the statement

\[x \text{ is even} \Rightarrow x^2 \text{ is even}.\]

And we’ve already proved that this is true, so we are done!