Is $R$ an equivalence relation or not?

Question

Let $\Bbb N := \{1,2,3,\dots\}$ and a relation is defined in $\Bbb N \times \Bbb N$ as follows. $(a, b)$ is related to $(c, d)$ if and only if $ad=bc$ then show whether $R$ is an equivalence relation or not.

Answer 1

$(a,b)R(c,d)\iff ad=bc$

reflexivity: $(a,b)R(a,b)$ proof: $ab=ba$

symmetry:

$(a,b)R(c,d)\iff (c,d)R(a,b)$

here: $ad=bc = cb=ad$.

transitivity: $(a,b)R((c,d)\land(c,d)R(e,f)\implies (a,b)R(e,f)$

$ad=bc \land cf=de\implies acf/e=bc\implies af=be$

$a,b,c,d,e,f\in N$ so none is 0.

$\square$