ThmDex – An index of mathematical definitions, results, and conjectures.
Result R5097 on D468: Bijective map
Power set is isomorphic to a set of boolean functions
Formulation 0
Let $X$ be a D11: Set such that
(i) $\mathcal{P}(X)$ is the D80: Power set of $X$
(ii) $\{ 0, 1 \}^X$ is the D4213: Set of Boolean functions from $X$ to $\{ 0, 1 \}$
Then \begin{equation} \mathcal{P}(X) \cong \{ 0, 1 \}^X \end{equation}
Proofs
Proof 0
Let $X$ be a D11: Set such that
(i) $\mathcal{P}(X)$ is the D80: Power set of $X$
(ii) $\{ 0, 1 \}^X$ is the D4213: Set of Boolean functions from $X$ to $\{ 0, 1 \}$
This result is a particular case of R1841: Indicator function operator is a bijection. $\square$