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Exercise 1: (i) Y∅ has exactly one element, namely ∅, whether Y is empty or not, and (ii) if X is not empty, then ∅ˣ is empty.

Exercise 2: Prove that (∪ᵢ Aᵢ) × (∪ⱼ Bⱼ) = ∪ᵢ,ⱼ (Aᵢ × Bⱼ) and that the same equation holds for intersections (provided that the domains of the families involved are not empty). Prove also (with appropriate provisos about empty families) that ∩ᵢ Xᵢ ⊂ Xⱼ ⊂ ∪ᵢ Xᵢ for each index j and that intersection and union can in fact be characterized as the extreme solutions of these inclusions. This means that if Xⱼ ⊂ Y for each index j, then ∪ᵢ Xᵢ ⊂ Y, and that ∪ᵢ Xᵢ is the only set satisfying the minimality condition; the formulation for intersections is similar.

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