Two-Valued Functions form Boolean Ring

Theorem

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

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

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

Then $\struct {2^S, +, \cdot}$ is a Boolean ring.

Proof

By Structure Induced by Ring Operations is Ring, $\struct {2^S, +, \cdot}$ is a ring.

By Unity of Induced Structure, $\struct {2^S, +, \cdot}$ also has a unity.

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

Hence $\struct {2^S, +, \cdot}$ is a Boolean ring.

$\blacksquare$

Sources

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