Question

Let $r$ be a relation over the set $N\times N$ and it is defined by $(a,b)r(c,d)\Rightarrow a+d=b+c$, then $r$ is:

• A. reflexive only
• B. symmetric only
• C. transitive only
• D. an equivalence relation

Answer 1

The correct option is D an equivalence relation

Assume $(a,b)r(a,b)$

$\Rightarrow a+b=a+b$ which is true

So, $r$ is reflexive.

Let $(a,b)r(c,d)$

$\Rightarrow a+d=b+c$

$\Rightarrow c+b=d+a$

$\Rightarrow (c,d)r(a,b)$.

So, $r$ is symmetric.

If $(a,b)r(c,d),(c,d)r(e,f)$ i.e if $a+d=b+c,c+f=d+e$ then $a+f=b+e$ and hence $(a,b)r(e,f)$.

This says the relation is transitive.

Hence, $r$ is an equivalence relation.