Question

If we define a relation $R$ on the set $N\times N$ as $(a,b)R(c,d)\iff a+d=b+c$ for all $(a,b),(c,d)ϵN\times N$, then the relation is

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

Answer 1

Answer: (C)

equivalence relation

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

$\Rightarrow R$ is reflexive.
Let $(a,b)R(c,d)\Rightarrow a+d=b+c$
$\Rightarrow c+b=d+a\Rightarrow (c,d)R(a,b)$

$\Rightarrow R$ is symmetric.
Next, Let $(a,b)R(c,d)$ and $(c,d)R(e,f)$
$\Rightarrow a+d=b+candc+f=d+e$
$\Rightarrow a+d+c+f=b+c+d+e$
$\Rightarrow a+f=b+e$ $\Rightarrow (a,b)R(e,f)$
$\Rightarrow R$ is transitive

$\Rightarrow R$ is an equivalence relation.