Chapter 4.1, Problem 30ES

In each of 28-31: a. Rewrite the theorem in three different ways: as $\forall$_______if_______then_______»
$\forall$_______________(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.

Theorm: The sum of any two odd integers is even.
Proof: Suppose m and n are any [particulasr but arbitarity chosen] odd integers.
[We must show that m+n is even.]
By (a) m+2r+1 and n=2s+1 for some integers r and s.

Theorem 4,1-2: The sum of any even integer and jjiv odd integer u odd.
Proof: Suppose m la 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 subsitution and slgebra.

$m+n=\left(c\right)=2\left(r+s\right)+1.$

Since r and s are both integers, so is their nun r + s. Hence m + n has the form twice some integer plus one and so (d)__by definition of odd.