Set Theory is Fun!

For any set $A$ , let $s(A)$ denote the number of subsets of A (this includes the empty set and $A$ itself). Suppose $X$ , $Y$ and $Z$ are sets such that both $X$ and $Y$ have 100 elements in them, and $X$ , $Y$ and $Z$ satisfy $s(X)+s(Y)+s(Z) = s(X or Y or Z)$ . Find the minimum number of elements in $X and Y and Z$ .

Clarification: By $X or Y or Z$ , I mean the union of the three sets, and by $X and Y and Z$ I mean the intersection of the three sets.

I got this problem from AMC.