IMC 2011 · Problem 4

Let $A_1, A_2, \dots, A_n$ be finite, nonempty sets. Define the function

$$f(t) = \sum_{k=1}^{n} \sum_{1 \le i_1 < i_2 < \dots < i_k \le n} (-1)^{k-1} t^{|A_{i_1} \cup A_{i_2} \cup \dots \cup A_{i_k}|}.$$

Prove that $f$ is nondecreasing on $[0,1]$.

($|A|$ denotes the number of elements in $A$.)