Question

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 $R=\left\{\right(x,y\left)|\right(x,y)\in H\times H$ and $x,y$ are faced in same direction$\}$. Then the relation '$R$' is

• A. Only symmetric and transitive
• B. Only reflexive and symmetric
• C. Only reflexive and transitive
• D. An equivalence relation

Answer 1

The correct option is D An equivalence relation

$x$ and $x$ are in the same direction.
Hence, $H$ is reflexive.

Let $(x,y)\in R\Rightarrow x$ and $y$ are in the same direction.
$\Rightarrow (y,x)\in H$
Hence, $H$ is symmetric.

Let $(x,y),(y,z)\in R\Rightarrow x$ and $y$ are in the same direction and $y$ and $z$ are in the same direction
Therefore, $x$ and $z$ are in the same direction
$\Rightarrow (x,z)\in H$
Hence, $H$ is transitive.

$H$ is an equivalance relation.