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:

$\powerset S \cap \powerset T = \powerset {S \cap T}$


Proof

\(\displaystyle X\) \(\in\) \(\displaystyle \powerset {S \cap T}\)
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle X\) \(\subseteq\) \(\displaystyle S \cap T\) Definition of Power Set
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle X\) \(\subseteq\) \(\displaystyle S \land X \subseteq T\) Definition of Set Intersection
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle X\) \(\in\) \(\displaystyle \powerset S \land X \in \powerset T\) Definition of Power Set
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle X\) \(\in\) \(\displaystyle \powerset S \cap \powerset T\) Definition of Set Intersection

$\blacksquare$


Also see


Sources