Let $A,B,C,D \subseteq X$ Show that $(A \setminus B ) \bigtriangleup (C \setminus D) \subseteq (A \bigtriangleup C) \cup (B \bigtriangleup D)$ Let $A,B,C,D \subseteq X$
Show that 
 $(A \setminus B ) \bigtriangleup (C \setminus D) \subseteq (A \bigtriangleup C) \cup (B \bigtriangleup D)$
My advances
$(A \setminus B ) \bigtriangleup (C \setminus D) $
$\rightarrow  \left [ (A \setminus B ) \cap (C \cap D^{c} )^{c} \right ] \cup$
 $\left [ (C \setminus D ) \cap ( A \cap B^{c} )^{c} \right ]   $
$\rightarrow \left [ (A \cap B^{c}) \cap (C \cap D^{c} )^{c} \right ] \cup$
   $\left [ (C \cap D ^{c}) \cap ( A \cap B^{c} )^{c} \right ]  $
$\rightarrow \left [ (A \cap B^{c}) \cap (C^{c} \cup D) \right] \cup \left [ (C \cap D ^{c}) \cap ( A^{c} \cup B ) \right ]   $
I can't conclude..
I Think, that is necessary find ... 
$(A \bigtriangleup C ) \cup (B \bigtriangleup D) $
$=\left [ (A \setminus C) \cup (C \setminus  A ) \right ] \cup \left [ (B \setminus  D) \cup (D  \setminus B ) \right ]   $
$\rightarrow \left [ (A \cap C^{c}) \cup (C \cap A ^{c}) \right ] \cup \left [ (B \cap D ^{c}) \cup ( D \cap B^{c}) \right ]   $
but I can't find their relationship..... I need help. Please!!!
 A: Another way to do it is to take an arbitrary element $x$ in the set on the left and show that it belongs to the set on the right. 
Consider the case $x \in A \setminus B, x \not\in C \setminus D$. Now $x \in C \cap D$ or $x \notin C$. In the first case, $x \in D \Delta B$ and in the second $x \in A \Delta C$. The case when $x \in C \setminus D$, $x \notin A \setminus B$ is symmetric; you can also perhaps convert this proof in algebraic terms.
A: \begin{align}
(A \setminus B ) \bigtriangleup (C \setminus D)&=((A \cap B^{c}) \cap (C \cap D^{c} )^{c})\cup ((C \cap D^{c}) \cap ( A \cap B^{c} )^{c})
\\
&=((A \cap B^{c}) \cap (C^{c} \cup D)) \cup ((C \cap D^{c}) \cap ( A^{c} \cup B ))
\\
&=(A \cap B^{c} \cap C^{c}) \cup (A \cap B^{c} \cap D) \cup (C \cap D^{c} \cap A^{c}) \cup (C \cap D^{c} \cap B)
\\
&\subset (A \cap C^{c}) \cup (B^{c} \cap D) \cup (C \cap A^{c}) \cup (D^{c} \cap B)\tag1
\\
&=(A \setminus C)\cup (D \setminus B)\cup (C \setminus A)\cup (B \setminus D)
\\
&=(A \bigtriangleup C) \cup (B \bigtriangleup D)
\end{align}
$(1)$ We have $A \cap B^{c} \cap C^{c}\subset A \cap C^{c}, \: A \cap B^{c} \cap D\subset B^{c} \cap D, \:C \cap D^{c} \cap A^{c}\subset C \cap A^{c}, \:C \cap D^{c} \cap B\subset D^{c} \cap B$
