Definition:Pointwise Inequality

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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.