Definition:Supremum
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Definition
Ordered Set
Let $\left({S, \preceq}\right)$ be a poset.
Let $T \subseteq S$.
An element $c \in S$ is the supremum of $T$ in $S$ if:
- $(1): \quad c$ is an upper bound of $T$ in $S$
- $(2): \quad c \preceq d$ for all upper bounds $d$ of $T$ in $S$.
The supremum of $T$ is denoted $\sup \left({T}\right)$.
The supremum of $x_1, x_2, \ldots, x_n$ is denoted $\sup \left\{{x_1, x_2, \ldots, x_n}\right\}$.
If there exists a supremum of $T$ (in $S$), we say that $T$ admits a supremum (in $S$).
The supremum of $T$ is often called the least upper bound of $T$ and denoted $\operatorname{lub} \left({T}\right)$.
Mapping
Let $f$ be a mapping defined on a subset of the real numbers $S \subseteq \R$.
Let $f$ be bounded above on $S$.
It follows from the Continuum Property that the codomain of $f$ has a supremum on $S$.
Thus:
- $\displaystyle \sup_{x \in S} f \left({x}\right) = \sup f \left({S}\right)$
Also defined 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.
Linguistic Note
The plural of supremum is suprema, although the (incorrect) form supremums can occasionally be found if you look hard enough.
