Question

$N$ is the set of natural numbers. The relation $R$ is defined on $N\times N$ as follow $(a,b)R(c,d)\iff a+d=b+c$. 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: we have

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

$\Rightarrow d+a=c+b$

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

R is transitive: let

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

Then by definition of $R$, we have

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

$\Rightarrow a+d+c+f=b+c+d+e$ or $a+f=b+e$.

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

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

$\Rightarrow (a,b)R(e,f)$