1
$\begingroup$

I am trying to prove that $S\cup(S\cap T)=S$ and the dual statement $S\cap(S\cup T)=S$ for a class, and have gotten stuck with my proof.

$$ S\cup(S\cap T) $$ $$ =\{x|x\in S\lor x\in (S\cap T)\} $$ $$ =\{x|x\in S\lor (x\in S\land ]x\in T)\} $$ $$ =\{x|(x\in S\lor x\in S)\land (x\in S\lor x\in T)\} $$

Any help on where to go from here would be appreciated.

  • 0
    A lot of times, when trying to show that two sets are equal, it is easier to show inclusion from both directions one step at a time. i.e If you want to show that $A=B$ then show that $A\subseteq B$ and that $B\subseteq A$. Take an arbitrary element from $A$ and show that it is in $B$ and then the other way round. I recommend you to try that before looking at the answers – 2017-02-20

1 Answers 1

0

We have

$$S\cup(S\cap T)$$

Let $R = S\cap T$

We know $R\subseteq S$ since the intersection of some set $S$ with any other set is a subset of $S$.

We also know the union of $S$ and any subset of $S$--including $R$--is $S$, i.e. $R\cup S=S$.

Therefore, $S\cup(S\cap T)=S$

  • 0
    I will need to prove the second statement as well to be able to use it. How would I do that? – 2017-02-21