Bourbaki

Exercise 5, § 5

Theory of Sets, II, p. 51

It is set in small type as supplementary.

Let (Xi)1≤i≤n(X_i)_{1 \leq i \leq n} be a finite family of sets. For every subset HH of the set of indices {1,n}\{1, n\}, let PH=⋃i∈HXiP_H = \bigcup_{i \in H} X_i and QH=⋂i∈HXiQ_H = \bigcap_{i \in H} X_i. Let Fk\mathfrak{F}_k be the set of subsets of {1,n}\{1, n\} having kk elements; now prove that one has:

⋃H∈FkQH⊃⋂H∈FkPHsik≤12(n+1) \bigcup_{H \in \mathfrak{F}_k} Q_H \supset \bigcap_{H \in \mathfrak{F}_k} P_H \quad \text{si} \quad k \leq \frac{1}{2}(n + 1)

and

⋃H∈FkQH⊂⋂H∈FkPHsik≥12(n+1) . \bigcup_{H \in \mathfrak{F}_k} Q_H \subset \bigcap_{H \in \mathfrak{F}_k} P_H \quad \text{si} \quad k \geq \frac{1}{2}(n + 1) \ .
Solution

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Machine translation of the English, by gpt-5-6. Not checked by a person.