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

Let’s refresh the concept of direct proof.

“A direct proof of a conditional statement p → q is constructed when the first step is the assumption that p is true; subsequent steps are constructed using rules of inference, with the final step showing that q must also be true. A direct proof shows that a conditional statement p → q is true by showing that if p is true, then q must also be true, so that the combination p true and q false never occurs.” Discrete Mathematics and its Applications by Rosen.

Let n be an even number.

n=2k for some integer k.

The additive inverse of n is -n.

If n=2k, then

-n=-2k= 2(-k)

Therefore, -n is also an even number.

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