Let H be the set of all houses in a village w | vedclass

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(D) relation $R$ is an equivalence relation if it is reflexive,symmetric,and transitive.$1$. Reflexivity: For any house $x \in H$,$x$ faces the same d

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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.$1$. Reflexivity: For any house $x \in H$,$x$ faces the same direction as itself. Thus,$(x, x) \in R$ for all $x \in H$. So,$R$ is reflexive.$2$. Symmetry: If $(x, y) \in R$,then $x$ and $y$ face the same direction. This implies that $y$ and $x$ also face the same direction. Thus,$(y, x) \in R$. So,$R$ is symmetric.$3$. Transitivity: If $(x, y) \in R$ and $(y, z) \in R$,then $x$ and $y$ face the same direction,and $y$ and $z$ face the same direction. This implies that $x$ and $z$ face the same direction. Thus,$(x, z) \in R$. So,$R$ is transitive.Since $R$ is reflexive,symmetric,and transitive,it is an equivalence relation.

Let $H$ be the set of all houses in a village where each house is faced in one of the directions,East,West,North,South. Let $R = \{ (x,y) | (x,y) \in H \times H \text{ and } x, y \text{ are faced in same direction} \}$. Then the relation $R$ is

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