Definition:Pointwise Minimum of Mappings

Definition
Let $X$ be a set, and let $\left({S, \preceq}\right)$ be a toset.

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

Let $\min$ be the min operation on $\left({S, \preceq}\right)$.

Then the pointwise minimum of $f$ and $g$, denoted $\min \left({f, g}\right)$, is defined by:


 * $\max \left({f, g}\right): X \to S: \min \left({f, g}\right) \, \left({x}\right) := \min \left({f \left({x}\right), g \left({x}\right)}\right)$

Pointwise minimum thence is an instance of a pointwise operation on mappings.

Examples

 * Pointwise Minimum of Extended Real-Valued Functions
 * Pointwise Minimum of Real-Valued Functions

Also see

 * Pointwise Maximum of Mappings, an analogous notion tied to the max operation
 * Operation Induced on Set of Mappings