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


Then:

$f + g = g + f$


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


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

$\blacksquare$