# set theory problem

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• Jan 28th 2009, 11:57 AM
ninano1205
set theory problem
(Headbang) How can we do this?
• Jan 28th 2009, 12:28 PM
Plato
$\displaystyle x \in \left[ {\bigcup\limits_\alpha {A_\alpha } \backslash \bigcup\limits_\alpha {B_\alpha } } \right]\; \Rightarrow \;x \in \left[ {\bigcup\limits_\alpha {A_\alpha } } \right] \wedge x \notin \left[ {\bigcup\limits_\alpha {B_\alpha } } \right]$
$\displaystyle x \in \left[ {\bigcup\limits_\alpha {A_\alpha } } \right] \wedge x \notin \left[ {\bigcup\limits_\alpha {B_\alpha } } \right]\; \Rightarrow \;\left( {\exists \gamma } \right)\left[ {x \in A_\gamma } \right] \wedge \left( {\forall \alpha } \right)\left[ {x \notin B_\alpha } \right]$
$\displaystyle \begin{gathered} \left( {\exists \gamma } \right)\left[ {x \in A_\gamma } \right] \wedge \left( {\forall \alpha } \right)\left[ {x \notin B_\alpha } \right]\; \Rightarrow \;x \in A_\gamma \backslash B_\gamma \hfill \\ x \in A_\gamma \backslash B_\gamma \; \Rightarrow \;x \in \left[ {\bigcup\limits_\alpha {A_\alpha \backslash B_\alpha } } \right]\; \hfill \\ \end{gathered}$