Let N be the set of natural numbers and R be the relation on 𝑵 × 𝑵 defines by (a, b) R (c, d) if ad=bc for all a, b, c, d ∈ 𝑵. Show that R is an equivalence relation.

Recall that a relation R is an equivalence relation if it is reflexive, symmetric and transitive.
Reflexive:
For any (a, b) ∈ N×N; ab=ba
$\Rightarrow (\mathrm{a},\mathrm{b})\mathrm{R}(\mathrm{a},\mathrm{b})$
Hence, R is reflexive.
Symmetric:
Assume (a, b)R(c, d) for any a, b, c, d ∈ N
$$\begin{gathered} \therefore \mathrm{ad}=\mathrm{bc} \\ \Rightarrow \mathrm{cd}=\mathrm{da} \\ \Rightarrow (\mathrm{c},\mathrm{d})\mathrm{R}(\mathrm{a},\mathrm{b}) \end{gathered}$$
Therefore, R is symmetric.
Transitive:
Assume (a, b)R(c, d) and (c, d)R(e, f) for a, b, c, d, e, f ∈ N
So ad=bc and ef=de
$$\begin{gathered} \Rightarrow \mathrm{adcf}=\mathrm{bcde} \\ \Rightarrow \mathrm{af}=\mathrm{be} \\ \Rightarrow (\mathrm{a},\mathrm{b})\mathrm{R}(\mathrm{e},\mathrm{f}) \end{gathered}$$
Hence, R is transitive.
Therefore, R an equivalence relation. Which we shown.