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