# Pointwise Multiplication on Integer-Valued Functions is Commutative

## 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:

$f \times g = g \times f$

## 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

$\blacksquare$