# Definition:Supremum of Set/Real Numbers

## 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 $\bigvee S$.

Some sources introduce the notation $\ds \sup_{y \mathop \in S} y$, 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:

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

$\sup S = 3$

### Example 2

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

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

$\sup T = 2$

### Example 3

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

$V := \set {x \in \R: x > 0}$

### 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.