Question

Let $R$ be a relation on the set of integers $Z$ given by $aRb\iff a=b\cdot 2^k$ for some integer $k.$ Then $R$ is

• A. reflexive but not symmetric relation
• B. reflexive and symmetric but not transitive relation
• C. symmetric and transitive but not reflexive relation
• D. an equivalence relation

Answer 1

The correct option is D an equivalence relation

Let $a\in Z$
Clearly, $aRa$ as $a=a\cdot 2^0$
$\therefore R$ is reflexive relation.

Let $a,b\in Z$ such that $aRb$
Then, $a=b\cdot 2^k$ for some integer $k$
$\Rightarrow b=a\cdot 2^{-k},$ where $-k\in Z$
$\Rightarrow bRa$
$\therefore R$ is symmetric relation.

Let $a,b,c\in Z$ such that $aRb$ and $bRc$
$\Rightarrow a=b\cdot 2^{k_1}$ and $b=c\cdot 2^{k_2}$ for some $k_1,k_2\in Z$
$\Rightarrow a=c\cdot 2^{k_1+k_2},$ where $k_1+k_2\in Z$
$\Rightarrow aRc$
$\therefore R$ is transitive relation.

Hence, $R$ is an equivalence relation on $Z$