Answer to Question #185249 in Discrete Mathematics for Evans

Question

Give a contrapositive proof of the theorem; "If n is an interfer and 3n + 2 is even, then n is even."?

Answer 1

Let us give a contrapositive proof of the theorem: "If $n$ is an interger and $3n + 2$ is even, then $n$ is even.".

For this suppose that $n$ is an interger and $3n + 2$ is even, but $n$ is not even, and hence $n$ is odd. Then $n=2k+1$ for some $k\in\mathbb Z.$ It follows that $3n+2=3(2k+1)+2=6k+3+2=(6k+4)+1=2(3k+2)+1$, where $3k+2$ is integer. Therefore, $3n+2$ is odd. We conclude that $3n+2$ is not even, that is we have a contradiction with initial assumption that $3n + 2$ is even.