# Two-Valued Functions form Boolean Ring

## Theorem

Let $X$ be a set, and let $2$ be the two ring.

Let $2^X$ be the set of all $2$-valued functions on $X$.

Denote with $+$ and $\cdot$ the operations induced on $2^X$ by $+_2$ and $\times_2$, respectively.

Then $\left({2^X, +, \cdot}\right)$ is a Boolean ring.

## Proof

By Structure Induced by Ring on Set of Mappings is Ring, $\left({2^X, +, \cdot}\right)$ is a ring.

By Unity of Induced Structure, $\left({2^X, +, \cdot}\right)$ also has a unity.

By Induced Structure Idempotent, $\cdot$ is an idempotent operation.

Hence $\left({2^X, +, \cdot}\right)$ is a Boolean ring.

$\blacksquare$

## Sources

- 2008: Paul Halmos and Steven Givant:
*Introduction to Boolean Algebras*... (previous) ... (next): $\S 1$: Exercise $3$