Given: $S=\{1,2,3,\dots ,10\}$ and  

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

For Reflexive, $M$ is subset of '$S$'

$\Rightarrow \varphi \in \mathrm{M}$ for $\varphi \cap \varphi =\varphi$

But it is given that relation is $\mathrm{A}\cap \mathrm{B}\ne \varphi$.

So, it is not reflexive.

For symmetric,

If ${}_{A}R_B\Rightarrow \mathrm{A}\cap \mathrm{B}\ne \varphi$, then ${}_{B}R_A\Rightarrow \mathrm{B}\cap \mathrm{A}\ne \mathrm{\varphi }$,

So, it is symmetric.

For transitive,

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

$\Rightarrow {}_{A}R_B, {}_{B}R_C$ is applicable but ${}_{A}R_C$ is not followed.

So, it not transitive

Hence, the given relation is symmetric only.