Question

Prove that the relation $R$ defined on set $Z$ as $aRb\iff a-b$ is divisible by $3$, is an equivalence relation.

Answer 1

$aRb\iff a-b$ divisible by $3$

Reflexive:

$aRa\iff a-a$ divisible by $3$..... True

Therefore, the given relation $R$ is a reflexive relation.

Symmetric:

$aRb\iff a-b$ divisible by $3\Rightarrow a-b=3k$, where $k$ is an integer.

Then,

$bRa\iff b-a$ divisible by $3$

Because $b-a=-(a-b)=-3k$ which is divisible by $3$.

Therefore, the given relation $R$ is a symmetric relation.

Transitive:

$aRb\iff a-b$ divisible by $3\Rightarrow a-b=3k$, where $k$ is an integer.

$bRc\iff b-c$ divisible by $3\Rightarrow b-c=3p$, where, $p$ is an integer

Then,

$aRc\iff a-c$ divisible by $3$

Because $a-c=(a-b)+(b-c)=3k+3p=3(k+p)$ which is divisible by $3$.

Therefore, the given relation $R$ is a transitive relation.

Since, the given relation $R$ satisfies the reflexive, symmetric, and transitive relation properties, Therefore, it is an equivalence relation.