Chapter 4.1, Problem 29ES

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: The negative of any even integer is even.
Proof: Suppose n is any [particular but arbitrarily chosen] even integer
[We must show that -n is even.]

By (a) n= 2k for some integer k,

Then

$\begin{array}{l}-n=-\left(2k\right)\text{by}\text{(b)}\\ \text{=2}\left(-k\right)\text{by}\text{algetra}\end{array}$
Let r = —k. Then r is an integer because (— 1) and k are integers and (c)
Hence —n = 2r, where r an integer, and so -n is even by (d) [as was to be shown]