Question

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

• A. Reflexive
• B. Symmetric
• C. Anti-symmetric
• D. Transitive

Answer 1

Answer: (B)

Symmetric

Let $A$ be a non-empty set in $X$.
$A\cap A=A\ne \varphi$
Hence, $A$ is not related to $A$.
Hence, R is not reflexive.
Let $A$ and $B$ be the disjoint sets in $X$
Let $ARB$
$\Rightarrow A\cap B=\varphi$
or, $B\cap A=\varphi$
$\Rightarrow BRA$
Hence, R is symmetric.
Let, $A,B,C\in X$

Let $ARB$ and $BRC$

$\Rightarrow A\cap B=\varphi ,B\cap C=\varphi$

It can happen that A and C may have a common element.

W can show by example $A=\{1,2,3,4\},B=\{5,6,7\},C=\{4,8\}$

Here, $A\cap B=\varphi ,B\cap C=\varphi$ but $A\cap C=\left\{4\right\}$
Hence, $R$ is not transitive.