Pointwise Multiplication on Integer-Valued Functions is Associative
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Theorem
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:
- $\paren {f \times g} \times h = f \times \paren {g \times h}$
That is, pointwise multiplication on integer-valued functions 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 of Integer-Valued Functions | ||||||||||
\(\ds \) | \(=\) | \(\ds \map f x \times \paren {\map g x \times \map h x}\) | Integer Multiplication is Associative | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\paren {f \times \paren {g \times h} } } x\) | Definition of Pointwise Multiplication of Integer-Valued Functions |
$\blacksquare$