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
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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
