Intersection of Power Sets

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Theorem

The intersection of the power sets of two sets $S$ and $T$ is equal to the power set of their intersection:

$\displaystyle \mathcal P \left({S}\right) \cap \mathcal P \left({T}\right) = \mathcal P \left({S \cap T}\right)$


Proof

\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle X\) \(\in\) \(\displaystyle \) \(\displaystyle \mathcal P \left({S \cap T}\right)\) \(\displaystyle \) \(\displaystyle \)                    
\(\displaystyle \) \(\displaystyle \iff\) \(\displaystyle \) \(\displaystyle X\) \(\subseteq\) \(\displaystyle \) \(\displaystyle S \cap T\) \(\displaystyle \) \(\displaystyle \)          Definition of Power Set          
\(\displaystyle \) \(\displaystyle \iff\) \(\displaystyle \) \(\displaystyle X\) \(\subseteq\) \(\displaystyle \) \(\displaystyle S \land X \subseteq T\) \(\displaystyle \) \(\displaystyle \)          Definition of intersection          
\(\displaystyle \) \(\displaystyle \iff\) \(\displaystyle \) \(\displaystyle X\) \(\in\) \(\displaystyle \) \(\displaystyle \mathcal P \left({S}\right) \land X \in \mathcal P \left({T}\right)\) \(\displaystyle \) \(\displaystyle \)          Definition of Power Set          
\(\displaystyle \) \(\displaystyle \iff\) \(\displaystyle \) \(\displaystyle X\) \(\in\) \(\displaystyle \) \(\displaystyle \mathcal P \left({S}\right) \cap \mathcal P \left({T}\right)\) \(\displaystyle \) \(\displaystyle \)          Definition of intersection          

$\blacksquare$


Sources