Power Set and Two-Valued Functions are Isomorphic Boolean Rings

Theorem

Let $S$ be a set.

Let $\mathbf 2$ be the Boolean ring two.

Let $\powerset S$ be the power set of $S$; by Power Set is Boolean Ring, it is a Boolean ring.

Let $\mathbf 2^S$ be the set of all mappings $f: S \to \mathbf 2$; by Two-Valued Functions form Boolean Ring, it is also a Boolean ring

Let $\chi_{\paren \cdot}: \powerset S \to \mathbf 2^S$ be the characteristic function operation.

Then $\chi_{\paren \cdot}$ is a ring isomorphism.

Proof

From Support Operation Inverse to Characteristic Function Operation, $\chi_{\paren \cdot}$ is a bijection.

It therefore suffices to establish it is a ring homomorphism.

By Characteristic Function of Symmetric Difference:

$\chi_{A * B} = \chi_A + \chi_B - 2 \chi_A \chi_B$

Since $\mathbf 2^S$ is a Boolean ring, by Idempotent Ring has Characteristic Two, the right hand side reduces to:

$\chi_{A * B} = \chi_A + \chi_B$

showing that $\chi_{\paren \cdot}$ preserves ring addition.

By Characteristic Function of Intersection: Variant 1:

$\chi_{A \cap B} = \chi_A \cdot \chi_B$

showing that $\chi_{\paren \cdot}$ preserves the ring product.

Hence $\chi_{\paren \cdot}$ is a ring homomorphism.

The result follows.

$\blacksquare$

Sources

• 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): $\S 3$