Question

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

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

Answer 1

The correct option is C

equivalence relation

Explanation for correct option:

Option $\left(C\right)$: equivalence relation

A relation on set $R$ on a set of $A$ is said to be an equivalence relation if it is reflexive, symmetric and transitive.

A relation $R$ on a set $A$ is Reflexive if $\left(a,a\right)\in R$

In given set of integer $a=2^k.b$ where $k$ is an integer,

Let's assume $k=0$ (an integer)

$a=2^k.b$

$\Rightarrow a=2^0.b$

$\Rightarrow a=b$

Thus, for every value will be in form of $\left(a,a\right)$, that means $R$ is reflexive.

A relation $R$ on a set $A$ is Symmetric if $\left(a,b\right)\in R\Rightarrow \left(b,a\right)\in R$

In a given set of integer $a=2^k.b$ where $k$ is an integer,

If $a=2^k.b$ then $b=2^{-k}a$ where $k,-k$ are integer.

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

Thus, $R$ is symmetric.

A relation $R$ on a set $A$ is transitive if $\left(a,b\right)\in R\left(b,c\right)\in R\left(a,c\right)\in R$

In a given set of integer $a=2^k.b$ where $k$ is an integer,

If $a=2^k.b$ and $b=2^{k}c$

then $c=2^k.a$.

where $2^k=\frac{a}{b}=\frac{b}{c}=\frac{c}{a}$ where $k$ is any integer.

Thus, $R$ is Transitive.

As, the relation $a=2^k.b$ is reflexive, symmetric, and transitive it means it is an equivalence relation.