Pointwise Addition on Real-Valued Functions is Commutative

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Definition

Let $S$ be a set.

Let $f, g: S \to \R$ be real-valued functions.

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


Then:

$f + g = g + f$


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


Proof

\(\, \displaystyle \forall x \in S: \, \) \(\displaystyle \map {\paren {f + g} } x\) \(=\) \(\displaystyle \map f x + \map g x\) Definition of Pointwise Addition of Real-Valued Functions
\(\displaystyle \) \(=\) \(\displaystyle \map g x + \map f x\) Real Addition is Commutative
\(\displaystyle \) \(=\) \(\displaystyle \map {\paren {g + f} } x\) Definition of Pointwise Addition of Real-Valued Functions

$\blacksquare$