Question

Let $X$ be a non-empty set and $P\left(X\right)$ be the set of all subsets of $X$. For $A,B\in P\left(X\right),$ ${}_A{R}_B$ if and only if $A\cap B=\varphi$ then the relation

• A. $R$ is not reflexive
• B. $R$ is symmetric
• C. $R$ is not transitive
• D. $R$ is an equivalence relation

Answer 1

Answer: (A), (B), (C)

The correct options are
A $R$ is not reflexive
B $R$ is symmetric
C $R$ is not transitive

$A,B$ $\in$ $P\left(x\right)$

$A$ $\cap$ $B=$ $\varphi$

$R$ is not reflexive since $(a,a$) $\notin$ $R$ for every $A$ $\in$ $X$ $(\because A\cap B=\varphi )$

$R$ is symmetric since $(a,b)$ $ϵ$ R and (b,a) $\in$ $R,a,b$ $\in$ $R$

$R$ is not transitive since $(a,b)$ $\in$ $R$ and $(b,c)$ $\in$ $R$ $\nRightarrow$ $(a,c)$ $\in$ $R$ $(\because A\cap B=\varphi )$