# Definition:Pointwise Multiplication 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 product of $f$ and $g$ is defined as:

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

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

## Also denoted as

Using the other common notational forms for multiplication, this definition can also be written:

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

or:

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