Minimal upper bounds

Finally, a partially ordered set may have many minimal upper bounds without having a least upper bound. Minimal upper bounds are those upper bounds for which there is no strictly smaller element that also is an upper bound. This does not say that each minimal upper bound is smaller than all other upper bounds, it merely is not greater. The distinction between "minimal" and "least" is only possible when the given order is not a total one. In a totally ordered set, like the real numbers, the concepts are the same.

As an example, let ${\displaystyle S}$ be the set of all finite subsets of natural numbers and consider the partially ordered set obtained by taking all sets from ${\displaystyle S}$ together with the set of integers ${\displaystyle \mathbb {Z} }$ and the set of positive real numbers ${\displaystyle \mathbb {R} ^{+},}$ ordered by subset inclusion as above. Then clearly both ${\displaystyle \mathbb {Z} }$ and ${\displaystyle \mathbb {R} ^{+}}$ are greater than all finite sets of natural numbers. Yet, neither is ${\displaystyle \mathbb {R} ^{+}}$ smaller than ${\displaystyle \mathbb {Z} }$ nor is the converse true: both sets are minimal upper bounds but none is a supremum.

Least-upper-bound property

The least-upper-bound property is an example of the aforementioned completeness properties which is typical for the set of real numbers. This property is sometimes called Dedekind completeness.

If an ordered set ${\displaystyle S}$ has the property that every nonempty subset of ${\displaystyle S}$ having an upper bound also has a least upper bound, then ${\displaystyle S}$ is said to have the least-upper-bound property. As noted above, the set ${\displaystyle \mathbb {R} }$ of all real numbers has the least-upper-bound property. Similarly, the set ${\displaystyle \mathbb {Z} }$ of integers has the least-upper-bound property; if ${\displaystyle S}$ is a nonempty subset of ${\displaystyle \mathbb {Z} }$ and there is some number ${\displaystyle n}$ such that every element ${\displaystyle s}$ of ${\displaystyle S}$ is less than or equal to ${\displaystyle n,}$ then there is a least upper bound ${\displaystyle u}$ for ${\displaystyle S,}$ an integer that is an upper bound for ${\displaystyle S}$ and is less than or equal to every other upper bound for ${\displaystyle S.}$ A well-ordered set also has the least-upper-bound property, and the empty subset has also a least upper bound: the minimum of the whole set.

An example of a set that lacks the least-upper-bound property is ${\displaystyle \mathbb {Q} ,}$ the set of rational numbers. Let ${\displaystyle S}$ be the set of all rational numbers ${\displaystyle q}$ such that ${\displaystyle q^{2}<2.}$ Then ${\displaystyle S}$ has an upper bound (${\displaystyle 1000,}$ for example, or ${\displaystyle 6}$) but no least upper bound in ${\displaystyle \mathbb {Q} }$: If we suppose ${\displaystyle p\in \mathbb {Q} }$ is the least upper bound, a contradiction is immediately deduced because between any two reals ${\displaystyle x}$ and ${\displaystyle y}$ (including ${\displaystyle {\sqrt {2}}}$ and ${\displaystyle p}$) there exists some rational ${\displaystyle r,}$ which itself would have to be the least upper bound (if ${\displaystyle p>{\sqrt {2}}}$) or a member of ${\displaystyle S}$ greater than ${\displaystyle p}$ (if ${\displaystyle p<{\sqrt {2}}}$). Another example is the hyperreals; there is no least upper bound of the set of positive infinitesimals.

There is a corresponding greatest-lower-bound property; an ordered set possesses the greatest-lower-bound property if and only if it also possesses the least-upper-bound property; the least-upper-bound of the set of lower bounds of a set is the greatest-lower-bound, and the greatest-lower-bound of the set of upper bounds of a set is the least-upper-bound of the set.

If in a partially ordered set ${\displaystyle P}$ every bounded subset has a supremum, this applies also, for any set ${\displaystyle X,}$ in the function space containing all functions from ${\displaystyle X}$ to ${\displaystyle P,}$ where ${\displaystyle f\leq g}$ if and only if ${\displaystyle f(x)\leq g(x)}$ for all ${\displaystyle x\in X.}$ For example, it applies for real functions, and, since these can be considered special cases of functions, for real ${\displaystyle n}$-tuples and sequences of real numbers.

The least-upper-bound property is an indicator of the suprema.

Properties

If ${\displaystyle A}$ is any set of real numbers then ${\displaystyle A\neq \varnothing }$ if and only if ${\displaystyle \sup A\geq \inf A,}$ and otherwise ${\displaystyle -\infty =\sup \varnothing <\inf \varnothing =\infty .}$^[2]

If ${\displaystyle A\subseteq B}$ are sets of real numbers then ${\displaystyle \inf A\geq \inf B}$ (unless ${\displaystyle A=\varnothing \neq B}$) and ${\displaystyle \sup A\leq \sup B.}$

Identifying infima and suprema

If the infimum of ${\displaystyle A}$ exists (that is, ${\displaystyle \inf A}$ is a real number) and if ${\displaystyle p}$ is any real number then ${\displaystyle p=\inf A}$ if and only if ${\displaystyle p}$ is a lower bound and for every ${\displaystyle \epsilon >0}$ there is an ${\displaystyle a_{\epsilon }\in A}$ with ${\displaystyle a_{\epsilon }<p+\epsilon .}$ Similarly, if ${\displaystyle \sup A}$ is a real number and if ${\displaystyle p}$ is any real number then ${\displaystyle p=\sup A}$ if and only if ${\displaystyle p}$ is an upper bound and if for every ${\displaystyle \epsilon >0}$ there is an ${\displaystyle a_{\epsilon }\in A}$ with ${\displaystyle a_{\epsilon }>p-\epsilon .}$

Relation to limits of sequences

If ${\displaystyle S\neq \varnothing }$ is any non-empty set of real numbers then there always exists a non-decreasing sequence ${\displaystyle s_{1}\leq s_{2}\leq \cdots }$ in ${\displaystyle S}$ such that ${\displaystyle \lim _{n\to \infty }s_{n}=\sup S.}$ Similarly, there will exist a (possibly different) non-increasing sequence ${\displaystyle s_{1}\geq s_{2}\geq \cdots }$ in ${\displaystyle S}$ such that ${\displaystyle \lim _{n\to \infty }s_{n}=\inf S.}$

Expressing the infimum and supremum as a limit of a such a sequence allows theorems from various branches of mathematics to be applied. Consider for example the well-known fact from topology that if ${\displaystyle f}$ is a continuous function and ${\displaystyle s_{1},s_{2},\ldots }$ is a sequence of points in its domain that converges to a point ${\displaystyle p,}$ then ${\displaystyle f\left(s_{1}\right),f\left(s_{2}\right),\ldots }$ necessarily converges to ${\displaystyle f(p).}$ It implies that if ${\displaystyle \lim _{n\to \infty }s_{n}=\sup S}$ is a real number (where all ${\displaystyle s_{1},s_{2},\ldots }$ are in ${\displaystyle S}$) and if ${\displaystyle f}$ is a continuous function whose domain contains ${\displaystyle S}$ and ${\displaystyle \sup S,}$ then

$${\displaystyle f(\sup S)=f\left(\lim _{n\to \infty }s_{n}\right)=\lim _{n\to \infty }f\left(s_{n}\right),}$$

which (for instance) guarantees^{[note 1]} that ${\displaystyle f(\sup S)}$ is an adherent point of the set ${\displaystyle f(S)\,{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\,\{f(s):s\in S\}.}$ If in addition to what has been assumed, the continuous function ${\displaystyle f}$ is also an increasing or non-decreasing function, then it is even possible to conclude that ${\displaystyle \sup f(S)=f(\sup S).}$ This may be applied, for instance, to conclude that whenever ${\displaystyle g}$ is a real (or complex) valued function with domain ${\displaystyle \Omega \neq \varnothing }$ whose sup norm ${\displaystyle \|g\|_{\infty }\,{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\,\sup _{x\in \Omega }|g(x)|}$ is finite, then for every non-negative real number ${\displaystyle q,}$

$${\displaystyle \|g\|_{\infty }^{q}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\left(\sup _{x\in \Omega }|g(x)|\right)^{q}=\sup _{x\in \Omega }\left(|g(x)|^{q}\right)}$$

since the map ${\displaystyle f:[0,\infty )\to \mathbb {R} }$ defined by ${\displaystyle f(x)=x^{q}}$ is a continuous non-decreasing function whose domain ${\displaystyle [0,\infty )}$ always contains ${\displaystyle S:=\{|g(x)|:x\in \Omega \}}$ and ${\displaystyle \sup S\,{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\,\|g\|_{\infty }.}$

Although this discussion focused on ${\displaystyle \sup ,}$ similar conclusions can be reached for ${\displaystyle \inf }$ with appropriate changes (such as requiring that ${\displaystyle f}$ be non-increasing rather than non-decreasing). Other norms defined in terms of ${\displaystyle \sup }$ or ${\displaystyle \inf }$ include the weak ${\displaystyle L^{p,w}}$ space norms (for ${\displaystyle 1\leq p<\infty }$), the norm on Lebesgue space ${\displaystyle L^{\infty }(\Omega ,\mu ),}$ and operator norms. Monotone sequences in ${\displaystyle S}$ that converge to ${\displaystyle \sup S}$ (or to ${\displaystyle \inf S}$) can also be used to help prove many of the formula given below, since addition and multiplication of real numbers are continuous operations.

Arithmetic operations on sets

The following formulas depend on a notation that conveniently generalizes arithmetic operations on sets. Throughout, ${\displaystyle A,B\subseteq \mathbb {R} }$ are sets of real numbers.

Sum of sets

The Minkowski sum of two sets ${\displaystyle A}$ and ${\displaystyle B}$ of real numbers is the set

$${\displaystyle A+B~:=~\{a+b:a\in A,b\in B\}}$$

consisting of all possible arithmetic sums of pairs of numbers, one from each set. The infimum and supremum of the Minkowski sum satisfies

$${\displaystyle \inf(A+B)=(\inf A)+(\inf B)}$$

and

$${\displaystyle \sup(A+B)=(\sup A)+(\sup B).}$$

Product of sets

The multiplication of two sets ${\displaystyle A}$ and ${\displaystyle B}$ of real numbers is defined similarly to their Minkowski sum:

$${\displaystyle A\cdot B~:=~\{a\cdot b:a\in A,b\in B\}.}$$

If ${\displaystyle A}$ and ${\displaystyle B}$ are nonempty sets of positive real numbers then ${\displaystyle \inf(A\cdot B)=(\inf A)\cdot (\inf B)}$ and similarly for suprema ${\displaystyle \sup(A\cdot B)=(\sup A)\cdot (\sup B).}$^[3]

Scalar product of a set

The product of a real number ${\displaystyle r}$ and a set ${\displaystyle B}$ of real numbers is the set

$${\displaystyle rB~:=~\{r\cdot b:b\in B\}.}$$

If ${\displaystyle r\geq 0}$ then

$${\displaystyle \inf(r\cdot A)=r(\inf A)\quad {\text{ and }}\quad \sup(r\cdot A)=r(\sup A),}$$

while if ${\displaystyle r\leq 0}$ then

$${\displaystyle \inf(r\cdot A)=r(\sup A)\quad {\text{ and }}\quad \sup(r\cdot A)=r(\inf A).}$$

Using ${\displaystyle r=-1}$ and the notation ${\textstyle -A:=(-1)A=\{-a:a\in A\},}$ it follows that

$${\displaystyle \inf(-A)=-\sup A\quad {\text{ and }}\quad \sup(-A)=-\inf A.}$$

Multiplicative inverse of a set

For any set ${\displaystyle S}$ that does not contain ${\displaystyle 0,}$ let

$${\displaystyle {\frac {1}{S}}~:=\;\left\{{\tfrac {1}{s}}:s\in S\right\}.}$$

If ${\displaystyle S\subseteq (0,\infty )}$ is non-empty then

$${\displaystyle {\frac {1}{\sup _{}S}}~=~\inf _{}{\frac {1}{S}}}$$

where this equation also holds when ${\displaystyle \sup _{}S=\infty }$ if the definition ${\displaystyle {\frac {1}{\infty }}:=0}$ is used.^{[note 2]} This equality may alternatively be written as ${\displaystyle {\frac {1}{\displaystyle \sup _{s\in S}s}}=\inf _{s\in S}{\tfrac {1}{s}}.}$ Moreover, ${\displaystyle \inf _{}S=0}$ if and only if ${\displaystyle \sup _{}{\tfrac {1}{S}}=\infty ,}$ where if^{[note 2]} ${\displaystyle \inf _{}S>0,}$ then ${\displaystyle {\tfrac {1}{\inf _{}S}}=\sup _{}{\tfrac {1}{S}}.}$

Infima

• The infimum of the set of numbers ${\displaystyle \{2,3,4\}}$ is ${\displaystyle 2.}$ The number ${\displaystyle 1}$ is a lower bound, but not the greatest lower bound, and hence not the infimum.
• More generally, if a set has a smallest element, then the smallest element is the infimum for the set. In this case, it is also called the minimum of the set.
• ${\displaystyle \inf\{1,2,3,\ldots \}=1.}$
• ${\displaystyle \inf\{x\in \mathbb {R} :0<x<1\}=0.}$
• ${\displaystyle \inf \left\{x\in \mathbb {Q} :x^{3}>2\right\}={\sqrt[{3}]{2}}.}$
• ${\displaystyle \inf \left\{(-1)^{n}+{\tfrac {1}{n}}:n=1,2,3,\ldots \right\}=-1.}$
• If ${\displaystyle \left(x_{n}\right)_{n=1}^{\infty }}$ is a decreasing sequence with limit ${\displaystyle x,}$ then ${\displaystyle \inf x_{n}=x.}$

Suprema

• The supremum of the set of numbers ${\displaystyle \{1,2,3\}}$ is ${\displaystyle 3.}$ The number ${\displaystyle 4}$ is an upper bound, but it is not the least upper bound, and hence is not the supremum.
• ${\displaystyle \sup\{x\in \mathbb {R} :0<x<1\}=\sup\{x\in \mathbb {R} :0\leq x\leq 1\}=1.}$
• ${\displaystyle \sup \left\{(-1)^{n}-{\tfrac {1}{n}}:n=1,2,3,\ldots \right\}=1.}$
• ${\displaystyle \sup\{a+b:a\in A,b\in B\}=\sup A+\sup B.}$
• ${\displaystyle \sup \left\{x\in \mathbb {Q} :x^{2}<2\right\}={\sqrt {2}}.}$

In the last example, the supremum of a set of rationals is irrational, which means that the rationals are incomplete.

One basic property of the supremum is

$${\displaystyle \sup\{f(t)+g(t):t\in A\}~\leq ~\sup\{f(t):t\in A\}+\sup\{g(t):t\in A\}}$$

for any functionals ${\displaystyle f}$ and ${\displaystyle g.}$

The supremum of a subset ${\displaystyle S}$ of ${\displaystyle (\mathbb {N} ,\mid \,)}$ where ${\displaystyle \,\mid \,}$ denotes "divides", is the lowest common multiple of the elements of ${\displaystyle S.}$

The supremum of a set ${\displaystyle S}$ containing subsets of some set ${\displaystyle X}$ is the union of the subsets when considering the partially ordered set ${\displaystyle (P(X),\subseteq )}$, where ${\displaystyle P}$ is the power set of ${\displaystyle X}$ and ${\displaystyle \,\subseteq \,}$ is subset.