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?

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A: Please see the below picture for detailed solution.

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A: PROOF :

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Q: 0 < a < b then <.

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A: You have written 2n=n+9, which cannot be done. Please see the proof below.

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.