a. Rewrite the following theorem in three different ways: as ∀ _____, if then, _____ as ∀ _____, _____ (without using the words if or then), and as If, _____ then _____ (without using an explicit universal quantifier).
b. Fill in the blanks in the proof of the theorem.
Theorem: The sum of any even integer and any odd integer is odd.
Proof: Suppose m is any even integer and n is (a). By definition of even, m = 2r for some (b), and by definition of odd, n = 2s + 1 for some integer s. By substitution and algebra,
m + n = (c) = 2(r + s) + 1.
Since r and s are both integers, so is their sum r + s. Hence m + n has the form twice some integer plus one, and so (d) by definition of odd.
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