Definition:Pointwise Inequality

Definition

Let $S$ be a set, and let $\struct {T, \preceq}$ 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:

$\forall s \in S: \map f s \preceq \map g s$

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

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Examples

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