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Where am I doing it wrong? Both of them should be simplified to $ A \cup (B \cap C)$

1$$ A \cup (B \cap (A \cup C) ) = A \cup (A \cup B^c)^c) \cap (A\cup C)$$ 2$$ A \cup ((B \cap A)\cup(B \cap C)) = A \cup (A^c \cup B)\cap (A\cup C)$$ 3$$ (A \cup (B \cap A )) \cup (A \cup (B \cap C)) = ((A \cup A^c )\cup B)\cap (A\cup C)$$ 4$$ A \cup (B \cap C) = B\cap (A\cup C)$$

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The normal argument is element chasing. That is assume $x$ is a member of the set on the left hand side and prove it is an element of the right hand side. Do the same assuming that $x$ is an element of the right hand side. If both hold the sides must be equal. – user45150 Feb 1 at 5:16

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Nevermind if solved it now. I was doing the De Morgans law in the wrong way. $A∪((B∩A)∪(B∩C))=A∪(A^c∩B)∩(A∪C)$

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