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How do you prove that (A ∪ B) ∩ (B ∪ C) ⊆ (A ∪ C)?


You don’t, because it’s false.

(A ∪ B) ∩ (B ∪ C)=B∪(A∩C)

Suppose that [A]=2 and [C]=3 and [B]=75

In that case [A ∪ B] ≤5, but B∪(A∩C)= (A ∪ B) ∩ (B ∪ C) ≥ 75, in which case

(A ∪ B) ∩ (B ∪ C)> (A ∪ C)

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