Definition:Supremum of Set/Real Numbers

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This page is about Supremum of Subset of Real Numbers. For other uses, see Supremum.

Definition

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 $\R$.


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

$T$ admits a supremum (in $\R$) or
$T$ has a supremum (in $\R$).


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


Definition by Propositional Function

Let $\family {a_j}_{j \mathop \in I}$ be a family of elements of the real numbers $\R$ indexed by $I$.

Let $\map R j$ be a propositional function of $j \in I$.


Then we can define the supremum of $\family {a_j}_{j \mathop \in I}$ as:

$\ds \sup_{\map R j} a_j := \text { the supremum of all $a_j$ such that $\map R j$ holds}$


If more than one propositional function is written under the supremum sign, they must all hold.


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 $\map {\mathrm {lub} } T$ or $\map {\mathrm {l.u.b.} } T$.

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 $\bigvee T$ or $\ds \bigvee_{y \mathop \in T} y$.


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


Some older sources, applying the concept to a (strictly) increasing real sequence, refer to a supremum as an upper limit.


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.


Examples

Example 1

The subset $S$ of the real numbers $\R$ defined as:

$S = \set {1, 2, 3}$

admits a supremum:

$\sup S = 3$


Example 2

The subset $T$ of the real numbers $\R$ defined as:

$T = \set {x \in \R: 1 \le x \le 2}$

admits a supremum:

$\sup T = 2$


Example 3

The subset $V$ of the real numbers $\R$ defined as:

$V := \set {x \in \R: x > 0}$

does not admit a supremum.


Example 4

Consider the set $A$ defined as:

$A = \set {3, 4}$

Then the supremum of $A$ is $4$.

However, $A$ contains no element $x$ such that:

$3 < x < 4$.


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

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