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HMMT February

Algebra Difficulty 5.0 AIME Find the answer

Find the number of pairs of union/intersection operations (□1,□2)∈{∪,∩}2\left(\square_{1}, \square_{2}\right) \in\{\cup, \cap\}^{2} satisfying the condition: for any sets S,TS, T, function f:S→Tf: S \rightarrow T, and subsets X,Y,ZX, Y, Z of SS, we have equality of sets f(X)□1(f(Y)□2f(Z))=f(X□1(Y□2Z))f(X) \square_{1}\left(f(Y) \square_{2} f(Z)\right)=f\left(X \square_{1}\left(Y \square_{2} Z\right)\right).

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

If and only if □1=□2=∪\square_{1}=\square_{2}=\cup. See http://math.stackexchange.com/questions/359693/overview-of-

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