The relation 'R' in N × N such that


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


(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.

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