Chapter 4.1, Problem 28ES

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) $m+\text{2}r+\text{1}$ and $n=2s+1$ for some integers r and s.
Then
   $\begin{array}{l}m+n=\left(2r+1\right)+\left(2s+1\right)by\text{(b)}\\ \text{=2}r+2s+2\\ =2\left(r+s+1\right)by\text{algerta}\end{array}$
Hence $m+n=2u$ where u is an integer, and so, by (d) m+n is even [as was to be shown].