Sets and Relations Part 1

Let S={1,2,3,...,10}
R={(A,B):A∩B≠ϕ;A,B∈M}
For Reflexive,
M is subset of ' S '
‌ So ‌ϕ∈M
for ϕ∩ϕ=ϕ
⇒ but relation is A∩B≠ϕ
So it is not reflexive.
For symmetric,
ARB ‌‌A∩B≠ϕ ,
⇒BRA‌‌⇒B∩A≠ϕ ,
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