Let $\sim$ be a relation on set $\mathbb{N}\times\mathbb{N}$ defined by $(x,y)\sim(z,w)$ if $xw=yz$. Prove that $\sim$ is an equivalence relation.

Question

I start by saying, let $(x,y)\in\mathbb{N}\times\mathbb{N}$. Since $xy=xy$ we have $(x,y)\sim (x,y)$ and $\sim$ is reflexive.

Let $(x,y),(z,w)\in\mathbb{N}\times\mathbb{N}$ and assume $(x,y)\sim(z,w)$. This means $xw=yz$. Since $yz=xw$, then we have that $(z,w)\sim (x,y)$ and $\sim$ is symmetric.

I’m not sure how to prove it is transitive.

Answer 1

Let $(x,y)\sim (z,w),\ (z,w)\sim (a,b)$. Then $xw=yz$ and $zb=aw$. So $xwzb=yzaw\Rightarrow xb=ay $. Which is $(x,y)\sim (a,b)\checkmark$