Question

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

• A. $x{R}_{1}y\iff |x|=|y|$
• B. $x{R}_{2}y\iff x\ge y$
• C. $x{R}_{3}y\iff x|y$
• D. $x{R}_{4}y\iff x<y$

Answer 1

The correct option is A $x{R}_{1}y\iff |x|=|y|$

Let's take option A,
$x{R}_{1}y\iff |x|=|y|$
Now, $|x|=|x|$
$\Rightarrow x{R}_{1}x$
Hence, $R_1$ is reflexive.
Now, let $x{R}_{1}y$
$\Rightarrow |x|=|y|$
or $|y|=|x|$
$\Rightarrow y{R}_{1}x$
Hence, $R_1$ is symmetric
Now, let $x{R}_{1}y,y{R}_{1}z$
$\Rightarrow |x|=|y|,|y|=|z|$
$\Rightarrow |x|=|z|$
$\Rightarrow x{R}_{z}x$
Hence, $R_1$ is transitive.
Hence, $R_1$ is an equivalence relation.
Clearly, $R_2$ is not an equivalence relation as $2\ge 3$ does not implies $3\ge 2$
Also, $R_3$ is not an equivalence relation as $x$ divides $y$, does not implies $y$ divides $x$
Also $R_4$ is not an equivalence relation because $2<3$ but $3>2$