Set Theory is Fun!

Discrete Mathematics Level pending

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.

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