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3

Part of 2018 IMAR Test

Problems(1)

IMAR Test 2018 P3

Source: IMAR Test 2018 P3

4/21/2021
Let SS be a finite set and let P(S)\mathcal{P}(S) be its power set, i.e., the set of all subsets of SS, the empty set and SS, inclusive. If A\mathcal{A} and B\mathcal{B} are non-empty subsets of P(S),\mathcal{P}(S), let A∨B={X:X⊆A∪B,A∈A,B∈B}.\mathcal{A}\vee \mathcal{B}=\{X:X\subseteq A\cup B,A\in\mathcal{A},B\in\mathcal{B}\}. Given a non-negative integer n⩽∣S∣,n\leqslant |S|, determine the minimal size A∨B\mathcal{A}\vee \mathcal{B} may have, where A\mathcal{A} and B\mathcal{B} are non-empty subsets of P(S)\mathcal{P}(S) such that ∣A∣+∣B∣>2n|\mathcal{A}|+|\mathcal{B}|>2^n.
Amer. Math. Monthly
combinatoricsSetsromania