proving a binary relation on $\mathbb{Z} × \mathbb{Z}$ is an equivalence relation.

Question

Define a binary relation ∼ on $\mathbb{Z}×\mathbb{Z}$ by $(a, b) ∼ (c, d)$ if and only if $ad = bc$.

Prove that ∼ is an equivalence relation.

I'm trying to prove symmetric of this relation. Do I just show that $ab=bc$ and $bc=ab$ are equivalent by commutative property? how can I prove it?

Answer 1

We have to show that $(a_1,a_2)\sim(b_1,b_2)\implies (b_1,b_2)\sim(a_1,a_2)$.

Now $$(a_1,a_2)\sim(b_1,b_2)\\\implies a_1b_2=a_2b_1\\\implies a_2b_1=a_1b_2\;\;\text{,since '=' is symmetric}\\\implies b_1a_2=b_2a_1\;\;\text{,since multiplication is commutative for reals}\\\implies (b_1,b_2)\sim(a_1,a_2)\,\,\square$$