How to show that ($A\cup B)\cap(A\cup C)$ is a subset of $A\cup B\cup C$? The problem 
  Prove $A\cup$$(B\cap$$C) \subseteq A\cup B\cup C $
My Work
  Suppose $x \in$ $A\cup$$(B\cap$$C)$
  By definition of union
   $x \in A$ or $x \in$ $(B\cap C)$, or both
  By definition of intersection 
   $x \in A$ or ($x \in B$ and $ x \in C)$ or both
  Symbolically 
   $x \in A$ $\lor$ ($x \in B$ $\land$ $ x \in C)$ 
  By Distributive Law of Logical Equivalences $p\lor (q \land r) \equiv (p \lor q) \land (p \lor r)$
  ($x \in A$ $\lor$ ($x \in B$ $\land$ $ x \in C)) \equiv$ ((x $\in A$ $\lor$ x $\in B$) $\land$ $( x \in A \lor x \in C))$
How can I go from saying  ($x \in A$ $\lor$ $x \in B$) $\land$ $( x \in A \lor x \in C)$ to saying $x \in A \cup  B  \cup C$ ?
 A: You have that $$(x\in A\lor x\in B)\wedge (x\in A\lor x\in C)\implies ((x\in A)\lor (x\in B))$$
You want to show that $$x\in A\cup B\cup C\iff (x\in A)\lor (x\in B)\lor (x\in C)\equiv$$
$$ \equiv ((x\in A)\lor (x\in B))\lor (x\in C)$$
A: There is an easier way to do this problem
Let at what you have here 
$x \in A$ or ($x \in B\land x \in C)$

Start reasoning from here. There are three possibilities. Let's discuss the first one
$x \in A$. If this is true, then you know that $x \in A\cup B\cup C$ because one part of the definition of union is "in A or in C or in C.
Let's discuss the second possibility
($x \in B\land x \in C)$
If we take a look at the definition of union again, it's basically saying in A or in the union of B and C. Now lets break down the definition of the union of B and C. That's basically saying in B, or in C, or in B and C. And woah lah, that's what you have right there.
Now let's discuss the last possibility, the and of the original  
$x \in A \land$ $x \in B\land x \in C$ 
Ie take a look at the definition of union of A, B, and C again - in A, in B, in C, or in A, B, and C. You again have something that matches up.
Therefore through all possible paths you can take when you have  $x \in A$ or ($x \in B\land x \in C)$, you endup as a member of $A\cup B\cup C$, you can say that the former is a subset of the latter.
