# Structure Induced by Commutative Operation is Commutative

## 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

$\blacksquare$