Pointwise Addition is Associative

Theorem

Let $S$ be a non-empty set.

Let $\mathbb F$ be one of the standard number sets: $\Z, \Q, \R$ or $\C$.

Let $f, g, h: S \to \mathbb F$ be functions.

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

Then:

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

That is, pointwise addition is associative.

Proof

\(\ds \forall x \in S: \, \)

\(\ds \map {\paren {\paren {f + g} + h} } x\)

\(=\)

\(\ds \paren {\map f x + \map g x} + \map h x\)

Definition of Pointwise Addition

\(\ds \)

\(=\)

\(\ds \map f x + \paren {\map g x + \map h x}\)

Associative Law of Addition

\(\ds \)

\(=\)

\(\ds \map {\paren {f + \paren {g + h} } } x\)

Definition of Pointwise Addition

$\blacksquare$

Specific Contexts

This result can be applied and proved in the context of the various standard number sets:

Pointwise Addition on Integer-Valued Functions is Associative

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:

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

Pointwise Addition on Rational-Valued Functions is Associative

Let $f, g, h: S \to \Q$ be rational-valued functions.

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

Then:

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

Pointwise Addition on Real-Valued Functions is Associative

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

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

Then:

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

Pointwise Addition on Complex-Valued Functions is Associative

Let $f, g, h: S \to \C$ be complex-valued functions.

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

Then:

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