No Bijection from a Set to its Power Set

Theorem

Let $S$ be a set, and let $\mathcal P \left({S}\right)$ be its power set.

There is no bijection $f: S \to \mathcal P \left({S}\right)$.

Proof

A bijection is by its definition also a surjection.

By Cantor's Theorem there is no surjection from $S$ to $\mathcal P \left({S}\right)$.

Hence the result.

$\blacksquare$

Sources

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