SectionB.14The Integers as Equivalence Classes of Ordered Pairs

Define a binary relation $\cong$ on the set $Z=\nonnegints \times\nonnegints$ by

Let $(a,b)\in Z\text{.}$ Then $a+b=b+a\text{,}$ so $(a,b)\cong(b,a)\text{.}$

Let $(a,b),(c,d)\in Z$ and suppose that $(a,b)\cong (c,d)\text{.}$ Then $a+d=b+c\text{,}$ so that $c+b=d+a\text{.}$ Thus $(c,d)\cong (a,b)\text{.}$

Let $(a,b), (c,d), (e,f)\in Z\text{.}$ Suppose that

Then $a+d=b+c$ and $c+f=d+e\text{.}$ Therefore,

\begin{equation*} (a+d)+(c+f) =(b+c)+(d+e). \end{equation*}

It follows that

\begin{equation*} (a+f)+(c+d) =(b+e)+(c+d). \end{equation*}

Thus $a+f = b+e$ so that $(a,b)\cong(e,f)\text{.}$

Now that we know that $\cong$ is an equivalence relation on $Z\text{,}$ we know that $\cong$ partitions $Z$ into equivalence classes. For an element $(a,b)\in Z\text{,}$ we denote the equivalence class of $(a,b)$ by $\langle (a,b)\rangle\text{.}$

Let $\ints$ denote the set of all equivalence classes of $Z$ determined by the equivalence relation $\cong\text{.}$ The elements of $\ints$ are called integers.