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f(A) = f(B) = max A\triangle B

Source: Romania IMO TST 1993 1.4

February 17, 2020
Subsetsfunctionalgebra

Problem Statement

Let YY be the family of all subsets of X={1,2,...,n}X = \{1,2,...,n\} (n>1n > 1) and let f:Y→Xf : Y \to X be an arbitrary mapping. Prove that there exist distinct subsets A,BA,B of XX such that f(A)=f(B)=maxA△Bf(A) = f(B) = max A\triangle B, where A△B=(A−B)∪(B−A)A\triangle B = (A-B)\cup(B-A).
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