Question

Let $R$ be a non-empty relation on a collection of sets defined by $ARB$ if and only if $A\cap B=\varphi$. Then, (pick the true statement)

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

Answer 1

The correct option is B $R$ is symmetric and not transitive

(i) Reflexive
$A\cap A=A\ne \varphi$
So; $(A,A)$ doesn't belongs to relation $R$,
$\therefore$ Relation $R$ is not reflexive.

(ii) Symmetric
If $A\cap B=\varphi$ then $B\cap A=\varphi$ is also true.
$\therefore$ Relation $R$ is not Symmetric relation.

(iii) Transitive
If $A\cap B=\varphi$ and $B\cap C=\varphi$, it need be true that $A\cap C=\varphi$
For example:
$A=\{1,2\},B=\{3,4\},C=\{1,5,6$}
$A\cap B=\varphi$ and $B\cap C=\varphi$ but
$A\cap C=\left\{1\right\}\ne \varphi$
$\therefore$ Relation $R$ is not transitive relation.