Definition:Pointwise Operation on Integer-Valued Functions
Definition
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:
Pointwise Addition
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.