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(D) relation $r$ is an equivalence relation if it is reflexive,symmetric,and transitive.$1$. Reflexive: For any $(a, b) \in N 	imes N$,we have $a + b

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An equivalence relation

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(D) relation $r$ is an equivalence relation if it is reflexive,symmetric,and transitive.$1$. Reflexive: For any $(a, b) \in N 	imes N$,we have $a + b = b + a$. Thus,$(a, b)r(a, b)$ holds. So,$r$ is reflexive.$2$. Symmetric: If $(a, b)r(c, d)$,then $a + d = b + c$. This implies $c + b = d + a$,which means $(c, d)r(a, b)$. So,$r$ is symmetric.$3$. Transitive: If $(a, b)r(c, d)$ and $(c, d)r(e, f)$,then $a + d = b + c$ and $c + f = d + e$. Adding these equations: $a + d + c + f = b + c + d + e$. Simplifying gives $a + f = b + e$,which means $(a, b)r(e, f)$. So,$r$ is transitive.Since $r$ is reflexive,symmetric,and transitive,it is an equivalence relation.

Let $r$ be a relation on the set $N \times N$ defined by $(a, b)r(c, d) \Rightarrow a + d = b + c$. Then $r$ is:

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