The relation R = {(x, y) : x, y in mathbb{Z} | vedclass

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(C) Given relation $R = \{(x, y) : x, y \in \mathbb{Z} 	ext{ and } x + y 	ext{ is even} \}$.$1$. Reflexivity: For any $x \in \mathbb{Z}$,$x + x = 2x

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an equivalence relation

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(C) Given relation $R = \{(x, y) : x, y \in \mathbb{Z} 	ext{ and } x + y 	ext{ is even} \}$.$1$. Reflexivity: For any $x \in \mathbb{Z}$,$x + x = 2x$,which is always an even integer. Thus,$(x, x) \in R$ for all $x \in \mathbb{Z}$. So,$R$ is reflexive.$2$. Symmetry: If $(x, y) \in R$,then $x + y$ is even. Since $x + y = y + x$,$y + x$ is also even. Thus,$(y, x) \in R$. So,$R$ is symmetric.$3$. Transitivity: If $(x, y) \in R$ and $(y, z) \in R$,then $x + y$ is even and $y + z$ is even. The sum of two even numbers is even,so $(x + y) + (y + z) = x + z + 2y$ is even. Since $2y$ is even,$x + z$ must be even. Thus,$(x, z) \in R$. So,$R$ is transitive.Since the relation is reflexive,symmetric,and transitive,it is an equivalence relation.

The relation $R = \{(x, y) : x, y \in \mathbb{Z} \text{ and } x + y \text{ is even} \}$ is :

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