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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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    I'm just wondering why did you say that this is propositional calculus. (Of course this can be equivalently be written using $\lor,\land,\lnot,\to$ but you did write it in set form)2011-09-26
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    @Asaf : my native language isn't english, so I did not know what to call it the first place, but I found this on google and thought this was it. I guess I was mistaken. What is it called really?2011-09-26
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    I considered this as an option; the other one was that you were taught this in some logic relation course. I have edited the title and tags.2011-09-26
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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

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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.