Definition:Supremum of Set

From ProofWiki
Jump to navigation Jump to search

This page is about Supremum in the context of Ordered Set. For other uses, see Supremum.

Definition

Let $\struct {S, \preceq}$ be an ordered set.

Let $T \subseteq S$.


An element $c \in S$ is the supremum of $T$ in $S$ if and only 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 $S$.


If there exists a supremum of $T$ (in $S$), we say that:

$T$ admits a supremum (in $S$) or
$T$ has a supremum (in $S$).


Finite Supremum

If $T$ is finite, $\sup T$ is called a finite supremum.


Subset of Real Numbers

The concept is usually encountered where $\struct {S, \preceq}$ is the set of real numbers under the usual ordering $\struct {\R, \le}$:


Let $T \subseteq \R$ be a subset of the real numbers.


A real number $c \in \R$ is the supremum of $T$ in $\R$ if and only if:

$(1): \quad c$ is an upper bound of $T$ in $\R$
$(2): \quad c \le d$ for all upper bounds $d$ of $T$ in $\R$.


The supremum of $T$ is denoted $\sup T$ or $\map \sup T$.


Also known as

Particularly in the field of analysis, the supremum of a set $T$ is often referred to as the least upper bound of $T$ and denoted $\map {\operatorname {lub} } T$ or $\map {\operatorname {l.u.b.} } T$.

Some sources refer to the supremum of a set as the supremum on a set.

Some sources refer to the supremum of a set as the join of the set and use the notation $\bigvee S$.


Some sources introduce the notation $\displaystyle \sup_{y \mathop \in S} y$, which may improve clarity in some circumstances.


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.


Also see

  • Results about suprema can be found here.


Linguistic Note

The plural of supremum is suprema, although the (incorrect) form supremums can occasionally be found if you look hard enough.


Sources