Question

N is the set of positive integers and $\sim$ be a relation on $N\times Ndefined(a,b)\sim (c,d)$ iff ad=bc.
Check the relation for being an equivalence relation.

• A. True
• B. False

Answer 1

The correct option is A True

$(a,b)\sim (c,d)$ iff $ad=bc$
Reflexivity:
We can write $ab=ab$
$\Rightarrow (a,b)\sim (a,b)$ (by given def)
Hence, $\sim$ is reflexive.
Symmetry:
Let $(a,b)\sim (c,d)$
$\Rightarrow ad=bc$
$\Rightarrow da=cb$ (Product of real numbers is commutative)
$orcb=da$
$\Rightarrow (c,d)\sim (a,b)$
Hence, $\sim$ is symmetric.
Transitivity:
Let $(a,b)\sim (c,d);(c,d)\sim (e,f)$
$\Rightarrow ad=bc;cf=de$
$\Rightarrow af=be$ (Since $\frac{d}{c}=\frac{f}{e}$)
$\Rightarrow (a,b)\sim (e,f)$
Hence, $\sim$ is transitive.
Hence, $\sim$ is an equivalence relation.