Definition:Supremum of Mapping

Definition
Let $S$ be a set.

Let $ \left({T, \preceq}\right)$ be an ordered set.

Let $f:S \to T$ be a mapping from $S$ to $T$.

Let $f \left[{S}\right]$, the image of $f$, admit a supremum.

Then the supremum of $f$ (on $S$) is defined by:
 * $\displaystyle \sup_{x \mathop \in S} f \left({x}\right) = \sup f \left[{S}\right]$

Also see

 * Definition:Infimum of Mapping