# Definition:Pointwise Addition of Integer-Valued Functions

## Definition

Let $S$ be a non-empty set.

Let $f, g: S \to \Z$ be integer-valued functions.

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

$f + g: S \to \Z:$
$\forall s \in S: \left({f + g}\right) \left({s}\right) := f \left({s}\right) + g \left({s}\right)$

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

Thus pointwise addition is seen to be an instance of a pointwise operation on integer-valued functions.