Question: Let X be a finite set, let 1 , 2 , ... , A 1 ,A 2 ,...,A n be subsets of X, and let

Let X be a finite set, let 1 , 2 , ... , A 1 ,A 2 ,...,A n be subsets of X, and let I be a subset of [ ] [n]. Show that the number of elements of X which belong to A i for all iI but for no other indices (i.e., xA i for all iI and x / A i for all [ ] \ i[n]\I) is [ ] ( 1 ) \ , IJ[n] (1) J\I A J , where the sum goes over all subsets J of [ ] [n] containing I, and = A J = jJ A j if J = and = A J =X if = J=

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