Pointwise Addition on Integer-Valued Functions is Associative

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Theorem

Let $S$ be a set.

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

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


Then:

$\paren {f + g} + h = f + \paren {g + h}$


That is, pointwise addition on integer-valued functions is associative.


Proof

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

$\blacksquare$