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A in F, A subset B subset S implies B in F

Source: VJIMC 2007 2.4

June 24, 2021
set theory

Problem Statement

Let SS be a finite set with n elements and F\mathcal F a family of subsets of SS with the following property: A∈F,A⊆B⊆S  ⟹  B∈F.A\in\mathcal F,A\subseteq B\subseteq S\implies B\in\mathcal F.Prove that the function f:[0,1]→Rf:[0,1]\to\mathbb R given by f(t):=∑A∈Ft∣A∣(1−t)∣S∖A∣f(t):=\sum_{A\in\mathcal F}t^{|A|}(1-t)^|S\setminus A|is nondecreasing (∣A∣|A| denotes the number of elements of AA).
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