A useful elementary correlation inequality (Q1114216)

Let Q be a finite set, and let Y be a random subset of Q such that the events \(\{\) \(y\in Y\}\) are independent as y goes through Q. Let \(X_ 1,X_ 2,...,X_ n\) be subsets of Q and set \(A_ j=\{X_ j\subset Y\}\). Assume that \(P(A_ j)\leq\). The inequalities \[ \prod^{n}_{j=1}P(A^ c_ j)\leq P(\cap^{n}_{j=1}A_ j)\leq [\prod^{n}_{j=1}P(A^ c_ j)] \exp \{2\sum_{(i,j)\in H}P(A_ i\cap A_ j)] \] are proved, where \(H=\{(i,j):X_ i\cap X_ j\neq \emptyset \}\). Applications are to appear elsewhere.