Proving $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ Let $A$, $B$ and $C$ be sets. Prove that $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$
I am not sure how to formally prove and in special write this intuitive simple question about sets.
Thank you
 A: Hint: TO prove two sets are equal, show they are subsets of each other.  So assume something is a member of the left hand side, and show it must be a member of the right hand side.  Then do the same in reverse.
A: You could say A n (B u C) is either true or false.
lets take the case when its false, well that implies omega not in A or omega not in (B u C)
for some omega in the set.
if omega not in A then the right side is always false which means false = false
if omega not in (B u C) then its not in either and therefore the right hand side is always false.
You can do the same for true. if the left hand side is true then omega is in A and it is in either B or C and for the right hand side to be true you only need it to be in two sets.
Write this out in a truth table and thats a sufficient proof.
sorry im not very good with latex.
A: When in doubt, element chase.  $M = N$ will be true if $x \in M \iff x \in N$.
If $x \in A\cap (B\cup C)$ then $x\in A$ and $x \in B\cup C$.  So $x \in B$ or $x \in C$.  So we have two possibilities i) either $x \in A$ and $x \in B$ so $x \in A\cap B$.  or ii) $x \in A$ and $x \in C$ so $x \in A\cap C$.
So either $x \in A\cap B$ or $x \in A\cap C$.  So $x \in (A\cap B)\cup (A\cap C)$.
So $x \in A\cap (B\cup C) \implies x \in (A\cap B)\cup (A\cap C)$.
Now if $x \in (A\cap B) \cup (A\cap C)$ then we have to options i) $x \in A\cap B$ so $x \in A$.  Or ii) $x \in A\cap C$ so $x \in A$. In either case we have $x \in  A$ so $x\in A$ is a certainty.
Now we either have i)$x\in A\cap B$ so $x \in B$ or we have ii) $x \in A\cap C$ so $x \in C$.  So we have either $x \in B$ or $x \in C$.  So $x \in B\cup C$.
So we have $x \in A$ and $x \in B\cup C$ so $x \in A\cap (B\cup C)$.
So $x\in (A\cap B)\cup (A\cap C) \implies  x \in A\cap (B\cup C)$
so $x \in A\cap (B\cup C) \iff x \in (A\cap B)\cup (A\cap C)$
So $A\cap (B\cup C)= (A\cap B)\cup (A\cap C)$.
