Two-Valued Functions form Boolean Ring

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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