Question

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

• A. An equivalence relation
• B. Reflexive but not symmetric
• C. Reflexive and transitive but not symmetric
• D. Reflexive and symmetric but not transitive

Answer 1

The correct option is A

An equivalence relation

Explanation for correct option:

Given, $R$ is a relation of the set and $a=2^k.b$ for some $k\in I$

Step 1: Check for reflexive relation

To show that $aRb$:

If $a\in R$

$$\begin{gathered} LHS=a \\ RHS=2^k.a \end{gathered}$$

for $k=0$

$LHS=RHS$

Therefore, $R$ is reflexive.

Step 2: Check for symmetry relation

If $\left(a,b\right)\in R$

$$\begin{gathered} \Rightarrow a=2^p.b \\ \Rightarrow \frac{a}{b}=2^p\left(1\right) \end{gathered}$$

To show $\left(b,a\right)\in R$

$$\begin{gathered} LHS=b \\ RHS=2^{k}a \end{gathered}$$

for $k=-p$

$$\begin{gathered} LHS=b \\ RHS=2^{-p}a \\ \Rightarrow LHS=RHS \end{gathered}$$ (by $\left(1\right)$)

Therefore$\left(b,a\right)\in R$

Therefore, $R$ is symmetric.

Step 3: Check for transitive relation

Let $\left(a,b\right)\in R$ and $\left(b,c\right)\in R$

$$\begin{gathered} \Rightarrow a=2^m.b \\ \Rightarrow \frac{a}{b}=2^m \end{gathered}$$

and

$$\begin{gathered} \Rightarrow b=2^n.c \\ \Rightarrow \frac{b}{c}=2^n \end{gathered}$$

then

$$\begin{gathered} \frac{a}{c}=\frac{{\displaystyle \frac{a}{b}}}{{\displaystyle \frac{c}{b}}}=\frac{2^m}{2^{-n}} \\ \Rightarrow a=2^{m+n}c \\ \Rightarrow \left(a,c\right)\in R \end{gathered}$$

Therefore, $R$ is transitive.

Hence, option (A) is the correct answer.