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I'm having troubles understanding how the element method works. I was given this question, and have a lot more like it, but just don't know where to start. This looks like a pretty easy one, and hopefully once I figure this one out, it'll sorta "click".

$$ (A \cap B)\cup C = (A \cup C) \cap B $$

I understand that I need to show that each side is a subset of the other, but I just don't know how to get it to that point. Any help would be really appreciated.

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This problem is false. The right hand side has at most $|B|$ elements while the left hand side has at least $|C|$ elements. –  lyj Feb 8 '13 at 4:53
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2 Answers 2

up vote 2 down vote accepted

Essentially, you want to show that each is the subset of the other, as you know.

To prove $(A\cap B)\cup C \subseteq (A \cup C)\cap B$, you start by supposing $$x \in (A \cap B)\cup C$$ and unpack what this means. The aim is to show that under this assumption, it must follow that $x \in (A \cup C)\cap B$.

So, let $x \in (A\cap B)\cup C$.

Then $x \in A\cap B$, or $x \in C$ (definition of union of sets).

This means $(x \in A$ and $x\in B$) or $x \in C$. By Demorgan's, we have $[x \in A$ or $x \in C]$ and $[(x \in B)$ or $x\in C]$. That is, $x \in (A \cup C)$ and ... etc. But there's a glitch: we can conclude $x \in (A \cup C)$,

but it does not follow that $x \in B$ or $x \in C$ implies $x \in B$. What we can show is that $$x \in (A\cup C)\cap(B \cup C)$$

And hence $$(A \cap B) \cup C \subseteq (A\cup C)\cap (B \cup C)$$


The problem is that the equivalence doesn't hold. Try to determine which inclusion (subset relation), if any, hold.* Can you find a counterexample to show the equality of the left hand side and right hand side fails to hold for all $A, B, C$?

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Thanks amWhy, but how can I manipulate it further? –  K. Barresi Feb 8 '13 at 4:58
    
Thanks a ton, I really appreciate the help –  K. Barresi Feb 8 '13 at 5:20
    
You're very welcome! –  amWhy Feb 8 '13 at 5:20
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Consider B that is disjoint from A and C. Then right hand side is empty but left hand side is C. Try to use Venn Diagram for intuition.

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