Power Set of Subset
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Theorem
Let $S \subseteq T$ where $S$ and $T$ are both sets.
Then:
- $\powerset S \subseteq \powerset T$
where $\powerset S$ denotes the power set of $S$.
Proof
\(\ds X\) | \(\in\) | \(\ds \powerset S\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds X\) | \(\subseteq\) | \(\ds S\) | Definition of Power Set | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds X\) | \(\subseteq\) | \(\ds T\) | as $S \subseteq T$: Subset Relation is Transitive | ||||||||||
\(\ds X\) | \(\in\) | \(\ds \powerset T\) | Definition of Power Set |
$\blacksquare$
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
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 2$: Sets and Subsets: Exercise $1 \ \text{(g)}$
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 6$ The power axiom: Exercise $6.1. \ \text {(c)}$