# Proof involving sets

• October 19th 2008, 11:01 AM
dolphinlover
Proof involving sets
X \ ( A ^ B ) = ( X \ A ) U ( X \ B )

I know I have to show (prove) containment in both directions to prove equality...but I'm not sure even where to start with this.

Thanx in advance for the help!
• October 19th 2008, 12:00 PM
Plato
It is easy.
$\begin{array}{rcl}
{X\backslash \left( {A \cap B} \right)} & \Leftrightarrow & {X \cap \left( {A \cap B} \right)^c } \\
{} & \Leftrightarrow & {X \cap \left( {A^c \cup B^c } \right)} \\
{} & \Leftrightarrow & {\left( {X \cap A^c } \right) \cup \left( {X \cap B^c } \right)} \\
{} & \Leftrightarrow & {\left( {X\backslash A} \right) \cup \left( {X\backslash B} \right)} \\

\end{array}$