TITLE: Constructing integers as equivalence classes of pairs of natural numbers
QUESTION [5 upvotes]: Can you tell me how to construct the integer numbers ($\mathbb Z$) as equivalence classes of pairs of natural numbers ($\mathbb N$)? And also tell me the commutative and associative law by an equivalence relation. Be sure to use only addition and multiplication.

REPLY [3 votes]: Yes, this is possible. Consider the relation $(a,b) \sim (c,d) \text{ iff } a+d = c+b$.
Verify that $\sim$ is an equivalence relation and let $[(a,b)]$ be the equivalence class of $(a,b)$ with respect to $\sim$. Define
$[(a,b)] +_\sim [(c,d)] := [(a+c,b+d)]$ and $-_\sim[(a,b)] := [(b,a)]$.
Prove that these are well-defined functions and that 
$$\pi \colon (\{ [(a,b)] \colon a,b \in \mathbb N \}, +_\sim) \to (\mathbb Z , +), [(a,b)] \mapsto a-b$$
is an group isomorphism.
I leave it to you to define $\cdot_\sim$ such that $\pi$ becomes a ring isomorphism.