a problem on the algebra of sets trying to prove that $A \cup ( A \cap B) = A $ for any set $A,B$. 
I am trying to use distributive law for sets, but keep coming to the same form. Is there a way to prove this ? 

Yes, the way to prove this is to prove $A \cup (A \cap B) \subseteq A$ and also that $A \subseteq A \cup (A \cap B)$.  So you have two statements to prove.  I'll help you with the first one and you can do the second one on your own.  
To prove $A \cup (A \cap B) \subseteq A$, we need to show if $x \in A \cup (A \cap B)$, then $x \in A$.
Well, let $x \in A \cup (A \cap B)$, then.  By definition of union, this means $x \in A$ or $x \in A \cap B$.  If $x \in A$, then we are done since that is what we wanted to show.  If on the other hand $x \in A \cap B$, then that means $x \in A$ and $x \in B$, so we get $x \in A$ in this case, too.  Thus, in every case, $x \in A$, so we are done.
Now it is up to you to prove if $x \in A$, then $x \in A \cup (A \cap B)$. 
