Question

On the set $Z$ of all integer numbers, define the relation $R$ by $aRb$ iff $a+2b$ is divided by $3$. Then the relation $R$ is

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

Answer 1

The correct option is D equivalence relation

If $a$ is an integer, then $a+2a=3a;$ which is divisible by $3$.
$\Rightarrow aRa\forall a\in Z$
$\therefore R$ is reflexive.

Let $aRb$ where $a,b\in Z$
$\Rightarrow a+2b=3k,$ where $k\in Z$
Now,
$b+2a=(3a+3b)-(a+2b)$
$=(3a+3b)-3k$
$=3(a+b-k)$
$\Rightarrow b+2a=3m,$ where $m\in Z$
So, $aRb\Rightarrow bRa$
$\therefore R$ is symmetric.

Again, let $aRb$ and $bRc$ where $a,b,c\in Z$
$\Rightarrow a+2b=3j$ and $b+2c=3k$
Now,
$a+2b=3j$
$a+2(3k-2c)=3j$
$a+2c=3j-6k+6c$
$a+2c=3(j-2k+2c)$
$a+2c=3m,$ where $m\in Z$ as $j,k,c\in Z$
So, $aRb,bRc\Rightarrow aRc$
$\therefore R$ is transitive.

Hence, $R$ is an equivalence relation.