InterviewSolution
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Let A, B and C be sets. Then show that A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) |
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Answer» We first show that A ∪ (B ∩ C) ⊂ (A ∪ B) ∩ (A ∪ C) Let x ∈ A ∪ (B ∩ C). Then x ∈ A or x ∈ B ∩ C ⇒ x ∈ A or (x ∈ B and x ∈ C) ⇒ (x ∈ A or x ∈ B) and (x ∈ A or x ∈ C) ⇒ (x ∈ A ∪ B) and (x ∈ A ∪ C) ⇒ x ∈ (A ∪ B) ∩ (A ∪ C) Thus, A ∪ (B ∩ C) ⊂ (A ∪ B) ∩ (A ∪ C) ... (1) Now we will show that (A ∪ B) ∩ (A ∪ C) ⊂ (A ∪ C) Let x ∈ (A ∪ B) ∩ (A ∪ C) ⇒ x ∈ A ∪ B and x ∈ A ∪ C ⇒ (x ∈ A or x ∈ B) and (x ∈ A or x ∈ C) ⇒ x ∈ A or (x ∈ B and x ∈ C) ⇒ x ∈ A or (x ∈ B ∩ C) ⇒ x ∈ A ∪ (B ∩ C) Thus, (A ∪ B) ∩ (A ∪ C) ⊂ A ∪ (B ∩ C) ... (2) So, from (1) and (2), we have A ∩ (B ∪ C) = (A ∪ B) ∩ (A ∪ C) |
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