Question

Let $R$ be a relation over the set $N\times N$ and its 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

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.