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

Answer: (D)

An equivalence relation

We have $(a,b)R(a,b)$ for all $(a,b)\in N\times N$. Since $a+b=b+a$.

Hence, $R$ is reflexive.

$R$ is symmetric for we have $(a,b)R(c,d)\Rightarrow a+d=b+c\Rightarrow d+a=b+c\Rightarrow c+b=d+a\Rightarrow (c,d)R(a,b)$
Hence $R$ is symmetric.

Then by definition of $R$, we have

$a+d=b+c$ and $c+f=d+e$

Hence by addition, we get

$a+d+c+f=b+c+d+e$ or $a+f=b+e$

Hence $(a,b)R(e,f)$

Thus $(a,b)R(c,d)$ and $(c,d)R(e,f)\Rightarrow (a,b)R(e,f)$

Hence $R$ is transitive.

Hence $R$ is an equivalence relation