Prove that if n is an integer and 3n + 2 is even, then n is even using
a) a proof by contraposition.
b) a proof by contradiction
a) Assume the negation of conclusion:
Let is an integer. Assume that is odd. By definition of odd numbers, exists integer such that
Hence
So by definition of odd numbers, is odd number. Thus we prove the negation of hypothesis.
Therefore we prove that if is an integer and is even, then is even using a proof by contraposition.
b)
Let is an integer. Assume that is even and is odd. Then by definition of odd numbers, there exists an integer such that Hence
So by definition of odd numbers, is odd number, that contradicts our assumption that is even.
Hence, it is not the case that is even and is odd.
Therefore we prove that if is an integer and is even, then is even using a proof by contradiction.
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