No Bijection from Set to its Power Set

Theorem

Let $S$ be a set.

Let $\powerset S$ denote the power set of $S$.

There is no bijection $f: S \to \powerset S$.

Proof

A bijection is by its definition also a surjection.

By Cantor's Theorem there is no surjection from $S$ to $\powerset S$.

Hence the result.

$\blacksquare$

Sources

• 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $2$. Set Theoretical Equivalence and Denumerability
• 1968: A.N. Kolmogorov and S.V. Fomin‎: Introductory Real Analysis: $\S 2.5$: Theorem $6$
• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: Mappings: $\S 14$