Direct Proof

Hi Community,

I am having some trouble with this question:

"Show that the additive inverse, or negative, of an even number is an even number using a direct proof. "

I know it's basically asking me to prove that the inverse of an even number is even. I'm having trouble expressing it in English/ Mathematical operators.

I have:

Let n = 2a, for all !n=2a, n is even

Someone help?

-M

The additive inverse of a number x is denoted -(x)

All even numbers are defined to be $\displaystyle \{ 2a | a\in\mathbb{Z} \}$

So we have an even number. Let's call it 2a.

Well it's inverse is $\displaystyle -(2a)=(-1)(2a)=(2a)(-1)=2(a(-1))=2(-a)$, but if $\displaystyle a\in\mathbb{Z}$, then $\displaystyle -a\in\mathbb{Z}$, and so $\displaystyle -(2a)$ is an even number