Power Set of Subset

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 $6.1. \ \text {(c)}$