# Definition:Pointwise Operation

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## Definition

Let $S$ be a set.

Let $\struct {T, \circ}$ be an algebraic structure.

Let $T^S$ be the set of all mappings from $S$ to $T$.

Let $f, g \in T^S$, that is, let $f: S \to T$ and $g: S \to T$ be mappings.

Then the operation $f \oplus g$ is defined on $T^S$ as follows:

- $f \oplus g: S \to T: \forall x \in S: \map {\paren {f \oplus g} } x = \map f x \circ \map g x$

The operation $\oplus$ is called the **pointwise operation on $T^S$ induced by $\circ$**.

### Induced Structure

The algebraic structure $\struct {T^S, \oplus}$ is called the **algebraic structure on $T^S$ induced by $\circ$**.

### Real-Valued Functions

Let $\R^S$ be the set of all mappings $f: S \to \R$, where $\R$ is the set of real numbers.

Let $\oplus$ be a binary operation on $\R$.

Define $\oplus: \R^S \times \R^S \to \R^S$, called **pointwise $\oplus$**, by:

- $\forall f, g \in \R^S: \forall s \in S: \map {\paren {f \oplus g} } s := \map f s \oplus \map g s$

In the above expression, the operator on the right hand side is the given $\oplus$ on the real numbers.

## Also known as

A **pointwise operation** is also referred to as an **induced operation**.

It is usual to use the same symbol for the **pointwise operation** as for the operation that induces it.

Thus one would refer to the **structure on $T^S$ induced by $\circ$** as $\struct {T^S, \circ}$.

In most reference works, the precise properties of a **pointwise operation** are taken to be implicitly inherited from its base operation.

## Examples

### Cube and Sine Functions

Let $f$ and $g$ be the real functions defined as

\(\ds \forall x \in \R: \, \) | \(\ds \map f x\) | \(=\) | \(\ds x^3\) | |||||||||||

\(\ds \forall x \in \R: \, \) | \(\ds \map g x\) | \(=\) | \(\ds \sin x\) |

Then for all $x \in \R$:

\(\ds \map {\paren {f + g} } x\) | \(=\) | \(\ds x^3 + \sin x\) | ||||||||||||

\(\ds \map {\paren {f - g} } x\) | \(=\) | \(\ds x^3 - \sin x\) | ||||||||||||

\(\ds \map {\paren {f \times g} } x = \map {\paren {g \times f} } x\) | \(=\) | \(\ds x^3 \sin x\) | ||||||||||||

\(\ds \map {\paren {f \times f} } x\) | \(=\) | \(\ds x^6\) |

Contrast this with:

\(\ds \map {\paren {f \circ g} } x\) | \(=\) | \(\ds \paren {\sin x}^3\) | ||||||||||||

\(\ds \map {\paren {g \circ f} } x\) | \(=\) | \(\ds \map \sin {x^3}\) | ||||||||||||

\(\ds \map {\paren {f \circ f} } x\) | \(=\) | \(\ds x^9\) |

## Also see

- Definition:Pointwise Operation on Number-Valued Functions: this definition crystallises when $T$ is taken to be one of the standard number sets $\N, \Z, \Q, \R$ and $\C$.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 13$: Compositions Induced on Cartesian Products and Function Spaces