Let S = 1, 2, 3, , 10 . Suppose M is the set of all the subsets of...

Given: $S = { 1, 2, 3, \ldots , 10 }$ and

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

For Reflexive, $M$ is subset of '$S$'

$⇒ ϕ \in M$ for $ϕ \cap ϕ = ϕ$

But it is given that relation is $A \cap B \ne ϕ$.

So, it is not reflexive.

For symmetric,

If $RB A ⇒ A \cap B \ne ϕ$, then $RA B ⇒ B \cap A \ne ϕ$,

So, it is symmetric.

For transitive,

If $A = { ( 1, 2 ) , ( 2, 3 ) }$, $B = { ( 2, 3 ) , ( 3, 4 ) }$ then $C = { ( 3, 4 ) , ( 5, 6 ) }$.

$⇒ RB A , RC B$ is applicable but $RC A$ is not followed.

So, it not transitive

Hence, the given relation is symmetric only.