Proving equivalence relation

Question

Let $Q$ be the following subset of $\mathbb{Z}\times \mathbb{Z}$:

$Q=\left \{ (a,b)\in \mathbb{Z}\times \mathbb{Z}: b\neq 0 \right \}$

Define the relation $\sim $ on $Q$ as

$(a,b)\sim (c,d)\Leftrightarrow ad=bc$

Proof that $\sim$ is an equivalence relation, and specify $[(2,3)]$ and more generally the equivalence class $[(a,b)]$. Try to give an explanation of $Q/\sim $

I know that an equivalence relation is reflexive, symmetric, and transitive. I am not sure on how to approach such a proof and then to specify the values.

Answer 1

• $\sim$ is reflexive: For all $(a,b) \in Q$ we have $ab = ba$, that is, $(a,b) \sim (a,b)$.
• $\sim$ is symmetric: For all $(a,b),(c,d) \in Q$ such that $(a,b) \sim (c,d)$ we have $ad=bc$ and then $cb = da$, that is, $(c,d) \sim (a,b)$.
• $\sim$ is transitive: For all $(a,b),(c,d),(e,f) \in Q$ such that $(a,b) \sim (c,d)$ and $(c,d) \sim (e,f)$ we have $ad=bc$ and $cf = de$. Thus, $$(af)\color{red}{d} = (ad)f = (bc)f = b(cf) = b(de) = (be)\color{red}{d}$$ and since $d\neq0$, it follows that $af = be$, that is, $(a,b) \sim (e,f)$.