Question

Which among the following relations on $Z$ is an equivalence relation

• A. $xRy\iff |x|=|y|$
• B. $xRy\iff x\ge y$
• C. $xRy\iff x>y$
• D. $xRy\iff x<y$

Answer 1

The correct option is A $xRy\iff |x|=|y|$

Let's consider $xRy\iff |x|=|y|$
Now, $|x|=|x|$
$\Rightarrow xRx$
Hence, $R$ is reflexive.
Now, let $xRy$ $\Rightarrow |x|=|y|$
or $|y|=|x|$ as equality commutative
$\Rightarrow yRx$
Hence, $R$ is symmetric.
Checking for transitive,
Let $(x,y)$ and $(y,z)$ satisfies $R$
Now, $xRy,yRz$
$\Rightarrow |x|=|y|,|y|=|z|$
$\Rightarrow |x|=|z|$
$\Rightarrow xRz$
$\therefore (x,z)$ satisfies $R$
Hence, $R$ is transitive.
Hence, $R$ is an equivalence relation.

$R$ is not symmetric relation as $3\ge 2$ does not implies $2\ge 3$. Hence $R$ is not an equivalence relation.

$R$ is not symmetric relation as $3>2$ does not implies $2>3$. Hence $R$ is not an equivalence relation.

$R$ is not symmetric relation as $1<3$ is true but $3<1$ is not true. Hence $R$ is not an equivalence relation.