# 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.

$\blacksquare$