Union of Power Sets
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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
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 $7 \ \text{(i)}$