Proof involving sets

**dolphinlover** · October 19th 2008, 11:01 AM

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!

**Plato** · October 19th 2008, 12:00 PM

It is easy.
$\begin{array}{rcl} {X\backslash \left( {A \cap B} \right)} & \Leftrightarrow & {X \cap \left( {A \cap B} \right)^c } \\ {} & \Leftrightarrow & {X \cap \left( {A^c \cup B^c } \right)} \\ {} & \Leftrightarrow & {\left( {X \cap A^c } \right) \cup \left( {X \cap B^c } \right)} \\ {} & \Leftrightarrow & {\left( {X\backslash A} \right) \cup \left( {X\backslash B} \right)} \\ \end{array}$

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