### Sets And Relations Question 4

#### Question 4 - 2024 (27 Jan Shift 1)

Let $S={1,2,3, \ldots, 10}$. Suppose $M$ is the set of all the subsets of $S$, then the relation $R={(A, B): A \cap B \neq \phi ; A, B \in M}$ is $:$

(1) symmetric and reflexive only

(2) reflexive only

(3) symmetric and transitive only

(4) symmetric only

## Show Answer

#### Answer (4)

#### Solution

Let $S={1,2,3, \ldots, 10}$

$R={(A, B): A \cap B \neq \phi ; A, B \in M}$

For Reflexive,

$M$ is subset of ’ $S$ '

So $\phi \in M$

for $\phi \cap \phi=\phi$

$\Rightarrow$ but relation is $A \cap B \neq \phi$

So it is not reflexive.

For symmetric,

$ARB \quad A \cap B \neq \phi$,

$\Rightarrow BRA \Rightarrow B \cap A \neq \phi$,

So it is symmetric.

For transitive,

If $A={(1,2),(2,3)}$

$B={(2,3),(3,4)}$

$C={(3,4),(5,6)}$

$ARB \& BRC$ but $A$ does not relate to $C$

So it not transitive