Proving Equivalence Relations and Quotient Sets

Question

Prove that the relation ∼ on $Z×Z$ given by $(a, b) ∼ (c, d)$ if $a+d = b+c$ is an equivalence relation. Give the quotient set $Z × Z/$ ∼.

Answer 1

Verifying that the relation is an equivalence relation is a matter of verifying $3$ properties, none difficult.

Let us for example verify transitivity, that if $(a,b)$ is equivalent to $(c,d)$, and $(c,d)$ is equivalent to $(e,f)$, then $(a,b)$ is equivalent to $(e,f)$.

So we know that (i) $a+d=b+c$ and (ii) $c+f=d+e$. We want to show that $a+f=b+e$.

From (1) and (ii) we get $a+d+c+f=b+c+d+e$. Now cancellation gives us what we want.

For the quotient set, we will prove that each equivalence class contains precisely one element of the shape $(x,0)$.

To show that every $(a,b)$ is equivalent to some $(x,0)$, note that $(a,b)$ is equivalent to $(a-b,0)$.

For the uniqueness part, suppose that $(x,0)$ is equivalent to $(y,0)$. Then $x+0=0+y$, so $x=y$.

Thus the equivalence classes can be identified with ordered pairs $(x,0)$. This in turn has a natural relationship with $x$. So the quotient structure can be identified with $\mathbb{Z}$.