 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2014/iii/paper-11/3/ii/solution

Codex  0 Created 2026-10-03 Updated 2026-10-06

α = 1_{A}

β = 1_{B}

γ = 1_{A \lor B}

and

δ = 1_{A \land B}

, where the union and intersection of set families are the collections of all pairwise unions and intersections.

α (A) β (B) = 1

, then

A \cup B \in A \lor B

and

A \cap B \in A \land B

, so the pointwise hypothesis holds. Otherwise its left side is zero. Summing these indicator functions gives

∣ A \lor B ∣ ∣ A \land B ∣ \geq ∣ A ∣ ∣ B ∣ .

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