Definition:Pointwise Inequality

Definition
Let $S$ be a set, and let $\left({T, \preceq}\right)$ be an ordered set.

Let $f, g: S \to T$ be mappings.

Then $f$ pointwise precedes $g$, denoted $f \preceq g$, :


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

Thence it can be seen that pointwise precedence is an instance of an induced relation on mappings.

Examples

 * Pointwise Inequality of Real-Valued Functions, where $T$ is taken to be $\R$ with its usual ordering
 * Pointwise Inequality of Extended Real-Valued Functions, where $T$ is taken to be the extended real numbers $\overline{\R}$ with their ordering