# Definition:Pointwise Inequality

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## 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 $f \preceq g$, if and only if:

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

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