MathDB

Let

S

be a set that contains

n

elements. Let

A_{1},A_{2},\cdots,A_{k}

k

distinct subsets of

S

, where k\geq 2, |A_{i}| \equal{} a_{i}\geq 1 ( 1\leq i\leq k). Prove that the number of subsets of

S

that don't contain any

A_{i} (1\leq i\leq k)

is greater than or equal to 2^n\prod_{i \equal{} 1}^k(1 \minus{} \frac {1}{2^{a_{i}}}).

Problem Statement