Pointwise Multiplication on Integer-Valued Functions is Commutative

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Definition

Let $S$ be a set.

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

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


Then:

$f \times g = g \times f$


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


Proof

\(\, \displaystyle \forall x \in S: \, \) \(\displaystyle \left({f \times g}\right) \left({x}\right)\) \(=\) \(\displaystyle f \left({x}\right) \times g \left({x}\right)\) Definition of Pointwise Multiplication
\(\displaystyle \) \(=\) \(\displaystyle g \left({x}\right) \times f \left({x}\right)\) Integer Multiplication is Commutative
\(\displaystyle \) \(=\) \(\displaystyle \left({g \times f}\right) \left({x}\right)\) Definition of Pointwise Multiplication

$\blacksquare$