Prove the following theorem: Theorem ∀n ∈ Z, n is either even or odd (but not...

Prove the following theorem:

Theorem

∀n ∈ Z, n is either even or odd (but not both).

Your proof must address the following points:

1. n is even or odd (and nothing else).

2. n is odd =⇒ n is not even (hint: contradiction).

3. n is even=⇒ n is not odd (hint: contrapositive).

The first point is a bit more difficult. Start by making a statement about 0. Then assuming that n is even, what can you say about n−1 and n+ 1? Likewise, assuming that n is odd, what can you say about n−1 and n+ 1. Can you organize these facts into an argument that shows that you have accounted for all possible n ∈ Z?

Expert Solution

Colby Messinger answered 2 months ago

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