MHB Proving Set Equality: {A ∪ (B ∩ C')} ∪ A ∪ C = A

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The discussion centers on proving the set equality {A ∪ (B ∩ C')} ∪ (A ∪ C) = A. It is argued that the statement is false using a specific universe and defined sets. With A = {1, 2, 3}, B = {1, 3, 4}, and C = {5}, the calculations show that the left side results in {1, 2, 3, 4, 5}, which does not equal A. Therefore, the conclusion is that the proposed set equality does not hold true. The proof demonstrates a clear contradiction to the original claim.
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Prove that {A UNION(B INTERSECT C')} UNION (A UNION C)=A
 
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You can't prove it, it's not true! Let the "universe" be {1, 2, 3, 4, 5}, A= {1, 2, 3}, B= {1, 3, 4}, and C= {5}. Then C' is {1, 2, 3, 4} which contains B so B union C' is {1, 2, 3, 4}. A union (B union C')= {1, 2, 3, 4}. A union C is {1, 2, 3, 5} so (A union (B union C')) union (A union C) is {1, 2, 3, 4, 5}, NOT A.
 

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