Definition:Pointwise Operation on Integer-Valued Functions

Definition

Let $S$ be a non-empty set.

Let $\Z^S$ be the set of all mappings $f: S \to \Z$, where $\Z$ is the set of integers.

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

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

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

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

Specific Instantiations

When $\oplus$ has a specific name, it is usual to name the corresponding pointwise operation by prepending pointwise to that name:

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

Then the pointwise sum of $f$ and $g$ is defined as:

$f + g: S \to \Z:$
$\forall s \in S: \map {\paren {f + g} } s := \map f s + \map g s$

where the $+$ on the right hand side is integer addition.

Pointwise Multiplication

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

Then the pointwise product of $f$ and $g$ is defined as:

$f \times g: S \to \Z:$
$\forall s \in S: \map {\paren {f \times g} } s := \map f s \times \map g s$

where the $\times$ on the right hand side is integer multiplication.

Also see

It can be seen that these definitions instantiate the more general Pointwise Operation on Number-Valued Functions.