MathDB
IMC Day 1 Problem 4

Source: combinaotrics/algebra [?]

August 1, 2011
functionprobabilityreal analysisreal analysis unsolved

Problem Statement

Let A1,A2,…,AnA_1,A_2,\dots, A_n be finite, nonempty sets. Define the function f(t)=∑k=1n∑1≤i1<i2<⋯<ik≤n(−1)k−1t∣Ai1∪Ai2∪⋯∪Aik∣.f(t)=\sum_{k=1}^n \sum_{1\leq i_1<i_2<\dots<i_k\leq n} (-1)^{k-1}t^{|A_{i_1}\cup A_{i_2}\cup \dots\cup A_{i_k}|}. Prove that ff is nondecreasing on [0,1].[0,1]. (∣A∣|A| denotes the number of elements in A.A.)
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