Structure Induced by Commutative Operation is Commutative

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Theorem

Let $\struct {T, \circ}$ be an algebraic structure, and let $S$ be a set.

Let $\struct {T^S, \oplus}$ be the structure on $T^S$ induced by $\circ$.

Let $\circ$ be a commutative operation.


Then the pointwise operation $\oplus$ induced on $T^S$ by $\circ$ is also commutative.


Proof

Let $\struct {T, \circ}$ be a commutative algebraic structure.

Let $f, g \in T^S$.


Then:

\(\ds \map {\paren {f \oplus g} } x\) \(=\) \(\ds \map f x \circ \map g x\) Definition of Pointwise Operation
\(\ds \) \(=\) \(\ds \map g x \circ \map f x\) $\circ$ is commutative
\(\ds \) \(=\) \(\ds \map {\paren {g \oplus f} } x\) Definition of Pointwise Operation

$\blacksquare$


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