Yes, vertex with 0 element is ϕ.
Let S is a set with n elements, S= {v1,v2,v3,,,,,,,,vn}
P(S) = { ϕ,{v1},{v2}...... {v1,v2,v3,,,,,,,,vn}} ,∣ P(S) ∣ = 2^{n}
E = Two vertices of G are adjacent if and only if the corresponding sets intersect in exactly two elements.
So our desired graph will be G = <P(S), E> .
So vertices of G will be vertex with 0 element ( ϕ) ,vertex with 1 element(v1,v2,....vn) so on.