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Archangel Macsika

Let A, B, and C be sets. Show that a) (A ∪ B) ⊆ (A ∪ B ∪ C). b) (A ∩ B ∩ C) ⊆ (A ∩ B). c) (B − A) ∪ (C − A) = (B ∪ C) − A.

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Let A, B, and C be sets.

a) Let us show that (A ∪ B) ⊆ (A ∪ B ∪ C).

Let x\in A ∪ B. Then x\in A or x\in B. It follows that x\in A or x\in B or x\in C, and hence x\in A ∪ B∪ C. We conclude that (A ∪ B) ⊆ (A ∪ B ∪ C).

b) Let us show that (A ∩ B ∩ C) ⊆ (A ∩ B).

Let x\in A \cap B \cap C. Then x\in A and x\in B and x\in C. It follows that x\in A and x\in B, and hence x\in A \cap B. We conclude that (A ∩ B ∩ C) ⊆ (A ∩ B).

c) Let us show that (B − A) ∪ (C − A) = (B ∪ C) − A.

Taking into account that X-Y=X\cap \overline{Y} and using distributive law, we conclude that

(B − A) ∪ (C − A) =(B\cap\overline{A})\cup(C\cap\overline{A}) =(B\cup C)\cap\overline{A} = (B ∪ C) − A.

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