The Integers

Every integer can be expressed as the difference between two natural numbers. In fact we introduce the integers $\mathbb{Z}$ as the set of equivalence classes under the following equivalence relation on $\ensuremath{\mathbb{N} }\times \ensuremath{\mathbb{N} }$:  

$$(a,b) \sim (c,d) \quad \textrm{if and only if} \quad a+d=b+c$$ (1)

Remark 2.1     Every integer x can be expressed in the form

$$x = \begin{cases} 2n, & \text{if $x$\space is \textit{even}} \\ 2n+1, & \text{if $x$\space is \textit{odd}} \end{cases}$$ (2)

for some integer n.

Theorem 2.2     The product of two odd integers is an odd integer.

 

Table 1: Products of Integers

product	even	odd
even	even	even
odd	even	odd

 

We leave it as an exercise for reader to prove that the product of an even integer with any integer is even and we summarize this in a Table 1.