Question

Let $\mathrm{X}$ be a family of sets and $R$ be a relation on $\mathrm{X}$ defined by $\mathrm{A}$ is disjoint from $\mathrm{B}$. Then $\mathrm{R}$ is

• A. Reflexive
• B. Symmetric
• C. Antisymmetric
• D. Transitive

Answer 1

The correct option is B

Symmetric

Explanation for the correct option(s):

Given,

Let $A$ be the non-empty set in $\mathrm{X}$. Then $\mathrm{R}=\mathrm{A}$ is disjoint from $\mathrm{B}$.

Condition $\left(\mathrm{i}\right)$ Reflexive:-

$\Rightarrow \mathrm{ARA}$

$\Rightarrow \mathrm{A}$is disjoint from $\mathrm{A}$

$\Rightarrow \mathrm{A}\cap \mathrm{A}\ne \mathrm{\varphi }$

Option (A) is not True.

Condition $\left(\mathrm{ii}\right)$ Symmetric:-

If $\mathrm{ARB}$ then $\mathrm{BRA}$

$\Rightarrow \mathrm{A}$ is disjoint from $\mathrm{B}$

$\Rightarrow \mathrm{B}$is disjoint from $\mathrm{A}$

$\Rightarrow \mathrm{A}\cap \mathrm{B}=\mathrm{\varphi }$ and $\mathrm{B}\cap \mathrm{A}=\mathrm{\varphi }$

Option (B) is True.

Condition $\left(\mathrm{iii}\right)$ Transitive:-

If $\mathrm{ARB}$ and $\mathrm{BRC}$ then $\mathrm{ARC}$

$\Rightarrow$ If $\mathrm{A}$ is disjoint from $\mathrm{B}$.

$\Rightarrow$ If $\mathrm{B}$ is disjoint from $\mathrm{C}$.

Then

$\Rightarrow \mathrm{A}$ is disjoint from $\mathrm{C}$.

$\Rightarrow \mathrm{A}\cap \mathrm{C}\ne 0$

Since $\mathrm{A}$ cannot be disjoint from $\mathrm{C}$ always.

Option (D) is not True.

Condition $\left(\mathrm{iv}\right)$ Antisymmetric:-

Since $\mathrm{ARB}$ is symmetric it can be inversed.

Therefore, $\mathrm{R}$ is Symmetric.

Hence, the correct answer is (B).