The relation 'R' in N $\times$ N such that (a, b) R (c, d) ⇔ a + d = b + c is ______________ .

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an equivalence relation We observe the following properties of relation R. Reflexivity: Let (a, b) ∈ N $\times$ N $\Rightarrow$ a, b ∈ N $\Rightarrow$ a+b = b+a $\Rightarrow$ (a, b) ∈ R So, R is reflexive on N$\times$N. Symmetry: Let (a, b), (c, d) ∈ N $\times$ N such that (a, b) R (c, d) $\Rightarrow$ a+d = b+c $\Rightarrow$ d+a = c +b $\Rightarrow$ (d, c), (b, a) ∈ R So, R is symmetric on N$\times$N. Transitivity : Let (a, b), (c, d), (e, f) ∈ 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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