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Hi I am learning group theory and encountered this: $$(B\cap (A\cap B)')\cup (B'\cap (A\cap B)) = B\cap (A'\cup B').$$

I don't understand how this is true, could someone please show me proof?

Thanks

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    Are $A$ and $B$ subgroups? What does $A'$ mean here? Derived subgroup?2011-05-25
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    What are $A$, $B$, etc.? Where is the union and intersection taking place?2011-05-25
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    A better word for "equals statement" is "equality".2011-05-25
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    Have you tried a truth table? It would be huge but I think it would be a good exercise in learning equivalence.2011-05-25

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Assuming the prime denotes complementation: The second term on the left, $B'\cap(A\cap B)$, is empty, since $B$ and $B'$ are disjoint. That leaves $B\cap(A\cap B)'$. You can use De Morgan's law $(A\cap B)'=A'\cup B'$ to transform this into the right-hand side.

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    @Jason, if this answers your question, please clarify it and retag it as (elementary-set-theory).2011-05-25
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    @Lhf: I have done it anyway. Jason, can put it back if needed.2011-05-25
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    @joriki: Aw so me thanks I was missing the disjoint definition :) btw how do you print the math formulas so nicely?2011-05-25
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    @Jason: Seems you've figured that one out in the meantime? :-)2011-05-25
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    I guys I have he he :)2011-05-25
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    @Jason: We use MathJax (a form of $\LaTeX$) to typeset our equations. A big manual is [here](http://www.math.harvard.edu/texman/). I learn by imitation, so I often look at how other people have formatted their answers by clicking the edit button next to their posts to see the 'raw code' behind the formatting.2011-05-28
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    @Michael: I used to look at the raw code by clicking the edit button, too, but then someone pointed out that you can actually right-click on any $\TeX$ output and select "Show Source" to see the code. (This shows everything except for the enclosing (single or double) dollar signs.)2011-05-28
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    Oh. That's quite nifty. Thanks for the tip! I will say though that, though I'm fairly decent at typesetting my problems sets, I never knew about `$$` and `\begin{align}` until I looked at some of the longer answers.2011-05-28