# The relation 'R' in N × N such that

Question:

The relation 'R' in N × N such that

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

(a) reflexive but not symmetric
(b) reflexive and transitive but not symmetric
(c) an equivalence relation
(d) none of the these

Solution:

(c) an equivalence relation

We observe the following properties of relation R.

Reflexivity: Let $(a, b) \in N \times N$

$\Rightarrow a, b \in N$

$\Rightarrow a+b=b+a$

$\Rightarrow(a, b) \in R$

So, $R$ is reflexive on $N \times N$.

Symmetry: Let $(a, b),(c, d) \in \mathrm{N} \times \mathrm{N}$ such that $(a, b) R(c, d)$

$\Rightarrow a+d=b+c$

$\Rightarrow d+a=c+b$

$\Rightarrow(d, c),(b, a) \in R$

So, $R$ is symmetric on $\mathrm{N} \times \mathrm{N}$.

Transitivity: Let $(a, b),(c, d),(e, f) \in N \times N$ such that $(a, b) R(c, d)$ and $(c, d) R(e, f)$

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

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

$\Rightarrow a+f=b+e$

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

So, $R$ is transitive on $N \times N$.

Hence, R is an equivalence relation on N.