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How to simplify $(E \cup F)\cap(F \cup G)$

My process:

$(E \cup F)\cap (F \cup G)$

$((E \cup F)\cap F) \cup ((E \cup F)\cap G)$

$F \cup (E\cap G) \cup (F\cap G)$

And this is where I get stuck. It should simplify to F ∪ EG, but I'm not sure how to turn $E\cap G ∪ F\cap G$ into $E\cap G$.

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    What work have you done on this problem? – 2017-02-03
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    Where, specifically, do you get stuck? Please post your work so far along with the question. – 2017-02-03
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    Well the original problem is to simplify (E ∪ F)(F ∪ G) so this has been my process: Distribute to get: ((E ∪ F)F) ∪ ((E ∪ F)G) And this is where the problem I posted begins: F ∪ (EG ∪ FG) – 2017-02-03
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    Please put the process *in the question* using edits. – 2017-02-03
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    Sorry, just edited it. – 2017-02-03
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    Lookup the [absorption laws](https://proofwiki.org/wiki/Absorption_Laws_(Set_Theory)/Union_with_Intersection). – 2017-02-03

2 Answers 2

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Hint: Commutation. Then what is $F\cup (F\cap G)~\cup (E\cap G)$?

In fact, do so right from the start.

$$\begin{align}&~(E\cup F)\cap (F\cup G)\\ =&~ && \text{Commutation}\\=&~F\cup (E\cap G) && \text{Distribution}\end{align}$$

Two east steps; just fill in the middle one and you are done.

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Don't you see how to simplify $F\cup FG$? Forget about $E$ for a moment and just draw a Venn diagram for $F$, $G$ and $F\cup FG$.