Union of Power Sets

Theorem

The union of the power sets of two sets $S$ and $T$ is a subset of the power set of their union:

$\powerset S \cup \powerset T \subseteq \powerset {S \cup T}$

Equality does not hold in general:

The union of the power sets of two sets $S$ and $T$ is not necessarily equal to the power set of their union.

$\ds X$

$\in$

$\ds \powerset S \cup \powerset T$

$\ds \leadsto \ \ $

$\ds X$

$\subseteq$

$\ds S \lor X \subseteq T$

Definition of Set Union and Definition of Power Set

$\ds \leadsto \ \ $

$\ds X$

$\subseteq$

$\ds S \cup T$

Definition of Set Union

$\ds \leadsto \ \ $

$\ds X$

$\in$

$\ds \powerset {S \cup T}$

Definition of Power Set

$\blacksquare$

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 $5 \ \text{(b)}$
1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Sets and Logic: Exercise $7 \ \text{(i)}$