Definition:Pointwise Operation on Rational-Valued Functions

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Definition

Let $S$ be a non-empty set.

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


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

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

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

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


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


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

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

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


Pointwise Multiplication

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


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

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

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


Also see

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