Pointwise Multiplication on Rational-Valued Functions is Commutative

Definition

Let $S$ be a non-empty set.

Let $f, g: S \to \Q$ be rational-valued functions.

Let $f \times g: S \to \Q$ denote the pointwise product of $f$ and $g$.

Then:

$f \times g = g \times f$

That is, pointwise multiplication of rational-valued functions is commutative.

Proof

\(\ds \forall x \in S: \, \)

\(\ds \map {\paren {f \times g} } x\)

\(=\)

\(\ds \map f x \times \map g x\)

Definition of Pointwise Multiplication of Rational-Valued Functions

\(\ds \)

\(=\)

\(\ds \map g x \times \map f x\)

Rational Multiplication is Commutative

\(\ds \)

\(=\)

\(\ds \map {\paren {g \times f} } x\)

Definition of Pointwise Multiplication of Rational-Valued Functions

$\blacksquare$