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:

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

Union of Power Sets not always Equal to Powerset of Union

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.

Proof

\(\ds X\)

\(\in\)

\(\ds \powerset S \cup \powerset T\)

\(\ds \leadsto \ \ \)

\(\ds X\)

\(\subseteq\)

\(\ds S \lor X \subseteq T\)

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

$\blacksquare$

Also see

• Intersection of Power Sets

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