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 $S$.
Plural: Suprema.
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:
- $\displaystyle \sup_{x \in S} f \left({x}\right) = \sup f \left({S}\right)$
Variants of Definition
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.
