Question

Show that the relation R in $N\times N$ defined by $(a,b)R(c,d)$ if $ad=bc$ is an equivalence relation.

Answer 1

$ad=bc$ in an equivalence relation.

Symmetric:

if $(a,b)R(c,d)\in N\times N$

$\Rightarrow ad=bc$

$\Rightarrow bc=ad$

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

$R$ is symmetric

Reflexive:

If $(a,b)\in N\times N$

$\Rightarrow ab=ab$

$(a,b)\in (a,b)$ . So $R$ is reflexive.

Transitive:

If $(a,b)R(c,d)$ and$(c,d)R(e,f)$

$\Rightarrow ad=bc$ and $cf=de$

$\Rightarrow \frac{a}{b}=\frac{c}{d}and\frac{c}{d}=\frac{e}{f}$

$\Rightarrow \frac{a}{b}=\frac{e}{f}$

$af=eb$

$(a,b)R(e,f)\in N\times N$

So $R$ is transitive

Therefore $R$ is in equivalence relation.