Given $\mathbb N=\{0,1,2,...\}$, the set of integers is defined by $$\mathbb Z=\Big\{\{(x,y)\mid (x,y)R(a,b)\}\mid (a,b)\in\mathbb N\times \mathbb N\Big\},$$ where $R$ is the binary relation defined by $$(a,b)R(c,d)\quad\Leftrightarrow\quad a+d=b+c.$$
When this construction is presented, usually the first step is to prove (or, at least to mention) that $R$ is an equivalence relation. The same is true about the construction of $\mathbb Q$ and $\mathbb R$.
Questions:
- Why do we have to know that the relations used to construct the numbers are equivalence relations?
- Are the reflexivity, symmetry, and transitivity really needed in the construction? Or are these properties just a coincidence?
Equivalence relations are a generalisation of the identity. Their properties are reflecting exactly that.
Now, why are we even using them? Well, we have a structure given to us, e.g. $\Bbb N$, and we want to define $\Bbb Z$ as the closure of $\Bbb N$ under additive inverses. We observe that we can think about an element of $\Bbb Z$ as the "difference between two natural numbers". Only that there is no difference operation, we only have $+$. So we want to understand how would the difference behave, given $+$.
We are smart, and we know what we're looking for, we want to identify differences, i.e. $a-b=c-d$. But we understand that $x-y=z$ should be the same as $x=z+y$, and so we get $a+d=c+b$. And this defines the equivalence relation on $\Bbb{N\times N}$ that we're looking for.
Now, because we were smart and used $+$ to define $-$, we can show that the operations of $+$ and $\cdot$, and the order $\leq$, all of which are defined in terms of $+$, will in fact extend to $\Bbb{N\times N}/{\sim}$, where $\sim$ is our equivalence relation, under the obvious embedding $n\mapsto[(n,0)]_{\sim}$.
And this understanding extends to $\Bbb{Q,R,C}$, and other type of constructions of this sort.