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$ A \cup (C \setminus A) \cup (A \cap B \cap C) = A \cup (C \cap \overline{A}) \cup (A \cap B \cap C) = A \cup C \cup (A \cap B \cap C) = ??. $

I have got this far, but have no idea now to continue.

I hope someone who is more seasoned in propositional calculus can help me.

Thanks in advance.

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    Well, since unions, intersection, and complements satisfy the same laws as $\vee$, $\wedge$, and $\neg$, namely the laws of Boolean algebra, we can call this propositional calculus, Boolean algebra, or unions and intersections.2011-09-27

1 Answers 1

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$A\cap B\cap C$ is a subset of $A$. Hence $A\cup(A\cap B\cap C)=A$. So $A\cup (C\setminus A)\cup(A\cap B\cap C)=A\cup(C\setminus A)=A\cup C$. You cannot reduce this further.