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 known as
Some sources refer to the supremum as being the upper bound. Using this convention, any element greater than this is not considered to be an upper bound.

Also see

 * Definition:Infimum of Mapping