Two-Valued Functions form Boolean Ring
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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 $3$