# From list to free group

Given natural numbers $\mathbb{N}$, we could construct integers $\mathbb{Z}$ by considering the equivalence relation on $\mathbb{N} \times \mathbb{N}$ as follows,

• $(a, b)\sim(c, d)$ if $a + d + r= b + c + r$ for some $r$

Obviously, it’s a equivalence relation from properties of commutativity. Thus we have equivalence class ${[} ( a , b ){]}$ where $a, b \in \mathbb{N}$. Then, ${[}(0, 0){]} = {[}(1, 1){]} = {[}(n , n){]}$, and $-{[}(a, b){]}={[}(b, a){]}$. This construction from a commutative monoid to abelian group is called Grothendieck group. Continue reading