TITLE: Proof of $A\cap(B\cup C) = (A\cap B)\cup(A\cap C)$
QUESTION [7 upvotes]: I have to resit a calculus exam and for some reason set proofs were never my best friend...
Anyway, on a practice exam I encountered the following proof:
$$A\cap(B\cup C) = (A\cap B)\cup(A\cap C)$$
When I draw a Venn-diagram it seems quite obvious but I couldn't manage to write the proof down properly.
If someone could help me, that'd be great!

REPLY [4 votes]: Hint: proofs by characteristic function :
for any $A \subset X$ define $1_A : X \to \{ 0,1\}$ by
$$1_A(x) = \left\{ 
\begin{eqnarray}
\begin{split}
1 & \mbox{if } x \in A \\
0 & \mbox{if } x \notin A \\
\end{split}
\end{eqnarray}\right.$$
then we have 

$A= B \Leftrightarrow 1_A = 1_B$
$1_{A \cap B}=1_A \cdot 1_B$
$1_{A \cup B}=1_A+1_B-1_A \cdot 1_B$

here we want prove$$ A\cap(B\cup C) = (A\cap B)\cup(A\cap C) \iff 1_{A\cap(B\cup C)} = 1_{(A\cap B)\cup(A\cap C)}$$
$\color{red}{proof:}$

1.$$ \large{1_{A\cap(B\cup C)}= 1_{A} 1_{B\cup C}=(1_{A} (1_B+1_C-1_B \cdot 1_C))=1_{A} 1_B+1_{A} 1_C-1_{A} \cdot1_B \cdot 1_C}$$
2.$$\large{1_{(A\cap B)\cup(A\cap C)}=1_{(A\cap B)}+1_{(A\cap C)}-1_{(A\cap B)}\cdot1_{(A\cap C)}=1_{A} 1_B+1_{A} 1_C-1_{A} \cdot1_B \cdot 1_C}$$