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
| \(\ds X\) | \(\in\) | \(\ds \powerset {S \cap T}\) | ||||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds X\) | \(\subseteq\) | \(\ds S \cap T\) | Definition of Power Set | ||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds X\) | \(\subseteq\) | \(\ds S \land X \subseteq T\) | Definition of Set Intersection | ||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds X\) | \(\in\) | \(\ds \powerset S \land X \in \powerset T\) | Definition of Power Set | ||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds X\) | \(\in\) | \(\ds \powerset S \cap \powerset T\) | 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 $7 \ \text{(ii)}$