## 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.