Pointwise Multiplication is Associative

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Theorem

Let $S$ be a non-empty set.

Let $\mathbb F$ be one of the standard number sets: $\Z, \Q, \R$ or $\C$.

Let $f, g, h: S \to \mathbb F$ be functions.

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


Then:

$\paren {f \times g} \times h = f \times \paren {g \times h}$


That is, pointwise multiplication is associative.


Proof

\(\ds \forall x \in S: \, \) \(\ds \map {\paren {\paren {f \times g} \times h} } x\) \(=\) \(\ds \paren {\map f x \times \map g x} \times \map h x\) Definition of Pointwise Multiplication
\(\ds \) \(=\) \(\ds \map f x \times \paren {\map g x \times \map h x}\) Associative Law of Multiplication
\(\ds \) \(=\) \(\ds \map {\paren {f \times \paren {g \times h} } } x\) Definition of Pointwise Multiplication

$\blacksquare$


Specific Contexts

This result can be applied and proved in the context of the various standard number sets:


Pointwise Multiplication on Integer-Valued Functions is Associative

Let $f, g, h: S \to \Z$ be integer-valued functions.

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


Then:

$\paren {f \times g} \times h = f \times \paren {g \times h}$


Pointwise Multiplication on Rational-Valued Functions is Associative

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

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


Then:

$\paren {f \times g} \times h = f \times \paren {g \times h}$


Pointwise Multiplication on Real-Valued Functions is Associative

Let $f, g, h: S \to \R$ be real-valued functions.

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


Then:

$\paren {f \times g} \times h = f \times \paren {g \times h}$


Pointwise Multiplication on Complex-Valued Functions is Associative

Let $f, g, h: S \to \C$ be complex-valued functions.

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


Then:

$\paren {f \times g} \times h = f \times \paren {g \times h}$