\begin{definition}[Definition:Pointwise Operation on Rational-Valued Functions]
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 {{RHS}} is the given $\oplus$ on the rational numbers.
\end{definition}