MHB What is the Identity for Unions and Intersections of Sets?

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The discussion focuses on proving the identity $(A\cup B)\cap (B\cup C)\cap (C\cup A) = (A\cap B)\cup (A\cap C)\cup (B\cap C)$. The proof involves demonstrating both subset relationships: first, showing that the left side is contained within the right side, and then vice versa. The argument starts with an element x in the left side, leading to the conclusion that x must belong to at least one of the intersections on the right side. The proof highlights the logical connections between the unions and intersections of the sets involved. Ultimately, the identity is established through careful reasoning about the relationships among the sets.
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$(A\cup B)\cap (B\cup C)\cap (C\cup A) = (A\cap B)\cup (A\cap C)\cup (B\cap C)$

For the identity, we will show $(A\cup B)\cap (B\cup C)\cap (C\cup A) \subseteq (A\cap B)\cup (A\cap C)\cup (B\cap C)$ and $(A\cup B)\cap (B\cup C)\cap (C\cup A) \supseteq (A\cap B)\cup (A\cap C)\cup (B\cap C)$.
Let $x\in (A\cup B)\cap (B\cup C)\cap (C\cup A)$.
Then $x\in A\cup B$ and $x\in B\cup C$ and $x\in C\cup A$.
So $x\in A$ or $x\in B$ and $x\in B$ or $x\in C$ and $x\in C$ or $x\in A$.

So I am stuck at this point.
 
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If ($x \in A$ or $x \in B$) and ($x \in B$ or $x \in C$) and ($x \in C$ or $x \in A$), then ($x \in A$ and $x \in B$) or ($x \in A$ and $x \in C$) or ($x \in B$ and $x \in C$). From there, $x \in (A \cap B) \cup (A \cap C) \cup (B \cap C)$.
 
I was reading documentation about the soundness and completeness of logic formal systems. Consider the following $$\vdash_S \phi$$ where ##S## is the proof-system making part the formal system and ##\phi## is a wff (well formed formula) of the formal language. Note the blank on left of the turnstile symbol ##\vdash_S##, as far as I can tell it actually represents the empty set. So what does it mean ? I guess it actually means ##\phi## is a theorem of the formal system, i.e. there is a...

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