# 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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