# Definition:Pointwise Minimum of Mappings

## Definition

Let $X$ be a set.

Let $\struct {S, \preceq}$ be a toset.

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

Let $\min$ be the min operation on $\struct {S, \preceq}$.

Then the pointwise minimum of $f$ and $g$, denoted $\map \min {f, g}$, is defined by:

$\map \min {f, g}: X \to S: \map {\map \min {f, g} } x := \map \min {\map f x, \map g x}$

Hence pointwise minimum is an instance of a pointwise operation on mappings.