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

[khoo]

Prove that {A UNION(B INTERSECT C')} UNION (A UNION C)=A

 

[HallsofIvy]

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.