Find the mistakes in the “proofs” shown.
Exercise
Theorem: The product of an even integer and an odd integer is even.
“Proof: Suppose m is an even integer and n is an odd integer. If m • n is even, then by definition of even there exists an integer r such that m • n = 2r. Also since m is even, there exists an integer p such that m = 2p, and since n is odd there exists an integer q such that n = 2q + 1. Thus
mn = (2p)(2q + 1) = 2r,
where r is an integer. By definition of even, then, m • n is even, as was to be shown.”
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