Definition:Pointwise Operation on Rational-Valued Functions

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 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:

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