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

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SUMMARY

The discussion centers on the proof of the set equality {A ∪ (B ∩ C')} ∪ A ∪ C = A, which is established as false. Using the universe {1, 2, 3, 4, 5}, with sets A = {1, 2, 3}, B = {1, 3, 4}, and C = {5}, the complement C' is determined to be {1, 2, 3, 4}. The union operations yield {1, 2, 3, 4} and {1, 2, 3, 5}, leading to the final union resulting in {1, 2, 3, 4, 5}, which does not equal A.

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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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