# Intersection of Power Sets

From ProofWiki

## 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

- Paul R. Halmos:
*Naive Set Theory*(1960)... (previous)... (next): $\S 5$: Complements and Powers - T.S. Blyth:
*Set Theory and Abstract Algebra*(1975)... (previous)... (next): $\S 2$: Exercise $5 \ \text{(a)}$ - Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*(1978)... (previous)... (next): Exercise $1.7 \ \text{(ii)}$