Question

Which one of the following relations on R is equivalence relation :

(A) $x R_1 y \Leftrightarrow|x|=|y|$

(B) $x R_2 y \Leftrightarrow x \geq y$

(C) $x R_3 y \Leftrightarrow x \mid y(x$ divides $y)$

(D) $x R_4 y \Leftrightarrow x<y$

Answer 1

SOLUTION : $\ x R_2 y \Leftrightarrow x \geq y$ is not symmetric relation

$x R_3 y \Leftrightarrow x / y \quad$ is not symmetric relation

$x R_4 y \Leftrightarrow x<y$ is not symmetric relation

$x R_1 y \Leftrightarrow|x|=|y|$ is reflexive, symmetric and transitive so equivalence relation