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!

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- Oct 19th 2008, 11:01 AMdolphinloverProof 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! - Oct 19th 2008, 12:00 PMPlato
It is easy.

$\displaystyle \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} $