# Pointwise Addition on Integer-Valued Functions is Commutative

## Definition

Let $S$ be a set.

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

Let $f + g: S \to \Z$ denote the pointwise sum of $f$ and $g$.

Then:

$f + g = g + f$

## Proof

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

$\blacksquare$