# Intersection of Power Sets

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## Contents

## 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

- 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 5$: Complements and Powers - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 2$. Sets of sets: Exercise $5 \ \text{(a)}$ - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): Chapter $1$: Sets and Logic: Exercise $7 \ \text{(ii)}$