For n∈Z, prove n^2 is odd if and only if n is odd.
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For , let us prove that is odd if and only if is odd.
Let be an odd number. Then for some Then where Therefore, is odd.
Let be an odd number. Let us prove that is odd using the method by contradiction. Suppose that is not odd, and hence is even, that is for some Then where It follows that is even. This contradiction proves that is odd.