Union of Intersections 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.

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