TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the infimum (abbreviated inf; plural infima) of a
subset In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

$S$ of a
partially ordered set upright=1.15, Fig.1 The Hasse diagram of the Power set, set of all subsets of a three-element set \, ordered by set inclusion, inclusion. Sets connected by an upward path, like \emptyset and \, are comparable, while e.g. \ and \ are not. In mathem ...
$P$ is a
greatest element In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined duality (order theory), dually, t ...
in $P$ that is less than or equal to all elements of $S,$ if such an element exists. Consequently, the term ''greatest lower bound'' (abbreviated as ) is also commonly used. The supremum (abbreviated sup; plural suprema) of a subset $S$ of a partially ordered set $P$ is the
least element In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
in $P$ that is greater than or equal to all elements of $S,$ if such an element exists. Consequently, the supremum is also referred to as the ''least upper bound'' (or ). The infimum is in a precise sense
dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality ** . . . see more cases in :Duality theories * Dual ...
to the concept of a supremum. Infima and suprema of
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s are common special cases that are important in
analysis Analysis is the process of breaking a complex topic or substance Substance may refer to: * Substance (Jainism), a term in Jain ontology to denote the base or owner of attributes * Chemical substance, a material with a definite chemical composit ...
, and especially in
Lebesgue integration In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
. However, the general definitions remain valid in the more abstract setting of
order theory Order theory is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geom ...
where arbitrary partially ordered sets are considered. The concepts of infimum and supremum are similar to
minimum In mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (math ...
and
maximum In mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (math ...

, but are more useful in analysis because they better characterize special sets which may have . For instance, the set of
positive real numbers In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
$\R^+$ (not including $0$) does not have a minimum, because any given element of $\R^+$ could simply be divided in half resulting in a smaller number that is still in $\R^+.$ There is, however, exactly one infimum of the positive real numbers: $0,$ which is smaller than all the positive real numbers and greater than any other real number which could be used as a lower bound.

# Formal definition

A of a subset $S$ of a partially ordered set $\left(P, \leq\right)$ is an element $a$ of $P$ such that * $a \leq x$ for all $x \in S.$ A lower bound $a$ of $S$ is called an (or , or ) of $S$ if * for all lower bounds $y$ of $S$ in $P,$ $y \leq a$ ($a$ is larger than or equal to any other lower bound). Similarly, an of a subset $S$ of a partially ordered set $\left(P, \leq\right)$ is an element $b$ of $P$ such that * $b \geq x$ for all $x \in S.$ An upper bound $b$ of $S$ is called a (or , or ) of $S$ if * for all upper bounds $z$ of $S$ in $P,$ $z \geq b$ ($b$ is less than or equal to any other upper bound).

# Existence and uniqueness

Infima and suprema do not necessarily exist. Existence of an infimum of a subset $S$ of $P$ can fail if $S$ has no lower bound at all, or if the set of lower bounds does not contain a greatest element. However, if an infimum or supremum does exist, it is unique. Consequently, partially ordered sets for which certain infima are known to exist become especially interesting. For instance, a lattice is a partially ordered set in which all subsets have both a supremum and an infimum, and a
complete lattice In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
is a partially ordered set in which subsets have both a supremum and an infimum. More information on the various classes of partially ordered sets that arise from such considerations are found in the article on completeness properties. If the supremum of a subset $S$ exists, it is unique. If $S$ contains a greatest element, then that element is the supremum; otherwise, the supremum does not belong to $S$ (or does not exist). Likewise, if the infimum exists, it is unique. If $S$ contains a least element, then that element is the infimum; otherwise, the infimum does not belong to $S$ (or does not exist).

# Relation to maximum and minimum elements

The infimum of a subset $S$ of a partially ordered set $P,$ assuming it exists, does not necessarily belong to $S.$ If it does, it is a minimum or least element of $S.$ Similarly, if the supremum of $S$ belongs to $S,$ it is a maximum or greatest element of $S.$ For example, consider the set of negative real numbers (excluding zero). This set has no greatest element, since for every element of the set, there is another, larger, element. For instance, for any negative real number $x,$ there is another negative real number $\tfrac,$ which is greater. On the other hand, every real number greater than or equal to zero is certainly an upper bound on this set. Hence, $0$ is the least upper bound of the negative reals, so the supremum is 0. This set has a supremum but no greatest element. However, the definition of maximal and minimal elements is more general. In particular, a set can have many maximal and minimal elements, whereas infima and suprema are unique. Whereas maxima and minima must be members of the subset that is under consideration, the infimum and supremum of a subset need not be members of that subset themselves.

## 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 $S$ be the set of all finite subsets of natural numbers and consider the partially ordered set obtained by taking all sets from $S$ together with the set of
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
s $\Z$ and the set of positive real numbers $\R^+,$ ordered by subset inclusion as above. Then clearly both $\Z$ and $\R^+$ are greater than all finite sets of natural numbers. Yet, neither is $\R^+$ smaller than $\Z$ nor is the converse true: both sets are minimal upper bounds but none is a supremum.

## Least-upper-bound property

The is an example of the aforementioned completeness properties which is typical for the set of real numbers. This property is sometimes called . If an ordered set $S$ has the property that every nonempty subset of $S$ having an upper bound also has a least upper bound, then $S$ is said to have the least-upper-bound property. As noted above, the set $\R$ of all real numbers has the least-upper-bound property. Similarly, the set $\Z$ of integers has the least-upper-bound property; if $S$ is a nonempty subset of $\Z$ and there is some number $n$ such that every element $s$ of $S$ is less than or equal to $n,$ then there is a least upper bound $u$ for $S,$ an integer that is an upper bound for $S$ and is less than or equal to every other upper bound for $S.$ A
well-order In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
ed 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 the least-upper-bound property is $\Q,$ the set of rational numbers. Let $S$ be the set of all rational numbers $q$ such that $q^2 < 2.$ Then $S$ has an upper bound ($1000,$ for example, or $6$) but no least upper bound in $\Q$: If we suppose $p \in \Q$ is the least upper bound, a contradiction is immediately deduced because between any two reals $x$ and $y$ (including $\sqrt$ and $p$) there exists some rational $p,$ which itself would have to be the least upper bound (if $p > \sqrt$) or a member of $S$ greater than $p$ (if $p < \sqrt$). Another example is the
hyperreals In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
; there is no least upper bound of the set of positive infinitesimals. There is a corresponding ; 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 $P$ every bounded subset has a supremum, this applies also, for any set $X,$ in the function space containing all functions from $X$ to $P,$ where $f \leq g$ if and only if $f\left(x\right) \leq g\left(x\right)$ for all $x \in X.$ For example, it applies for real functions, and, since these can be considered special cases of functions, for real $n$-tuples and sequences of real numbers. The
least-upper-bound property In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
is an indicator of the suprema.

# Infima and suprema of real numbers

In
analysis Analysis is the process of breaking a complex topic or substance Substance may refer to: * Substance (Jainism), a term in Jain ontology to denote the base or owner of attributes * Chemical substance, a material with a definite chemical composit ...
, infima and suprema of subsets $S$ of the
real numbers In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

are particularly important. For instance, the negative
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s do not have a greatest element, and their supremum is $0$ (which is not a negative real number). The
completeness of the real numbers Intuitively, completeness implies that there are not any “gaps” (in Dedekind's terminology) or “missing points” in the real number line In mathematics, the real line, or real number line is the line (geometry), line whose Point (geometr ...
implies (and is equivalent to) that any bounded nonempty subset $S$ of the real numbers has an infimum and a supremum. If $S$ is not bounded below, one often formally writes $\inf_ S = -\infty.$ If $S$ is , one writes $\inf_ S = +\infty.$

## Properties

The following formulas depend on a notation that conveniently generalizes arithmetic operations on sets: Let the sets $A, B \subseteq \R,$ and scalar $r \in \R.$ Define * $A \neq \varnothing$ if and only if $\sup A \geq \inf A,$ and otherwise $-\infty = \sup \varnothing < \inf \varnothing = \infty.$ * $r A = \$; the scalar product of a set is just the scalar multiplied by every element in the set. * $A + B = \$; called the
Minkowski sum In geometry, the Minkowski sum (also known as Dilation (morphology), dilation) of two set (mathematics), sets of position vectors ''A'' and ''B'' in Euclidean space is formed by adding each vector in ''A'' to each vector in ''B'', i.e., the set : ...

, it is the arithmetic sum of two sets is the sum of all possible pairs of numbers, one from each set. * $A \cdot B = \$; the arithmetic product of two sets is all products of pairs of elements, one from each set. * If $\varnothing \neq S \subseteq \R$ then there exists a sequence $s_ = \left\left(s_n\right\right)_^$ in $S$ such that $\lim_ s_n = \sup S.$ Similarly, there will exist a (possibly different) sequence $s_$ in $S$ such that $\lim_ s_n = \inf S.$ Consequently, if the limit $\lim_ s_n = \sup S$ is a real number and if $f : \R \to X$ is a continuous function, then $f\left\left(\sup S\right\right)$ is necessarily an
adherent point In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
of $f\left(S\right).$ In those cases where the infima and suprema of the sets $A$ and $B$ exist, the following identities hold: * $p = \inf A$ if and only $p$ is a lower bound and for every $\epsilon > 0$ there is an $a_\epsilon \in A$ with $a_\epsilon < p + \epsilon.$ * $p = \sup A$ if and only $p$ is an upper bound and if for every $\epsilon > 0$ there is an $a_\epsilon \in A$ with $a_\epsilon > p - \epsilon$ * If $A \subseteq B$ and then $\inf A \geq \inf B$ and $\sup A \leq \sup B.$ * If $r > 0$ then $\inf \left(r \cdot A\right) = r \left\left(\inf A\right\right)$ and $\sup \left(r \cdot A\right) = r \left\left(\sup A\right\right).$ * If $r \leq 0$ then $\inf \left(r \cdot A\right) = r \left\left(\sup A\right\right)$ and $\sup \left(r \cdot A\right) = r \left\left(\inf A\right\right).$ * $\inf \left(A + B\right) = \left\left(\inf A\right\right) + \left\left(\inf B\right\right)$ and $\sup \left(A + B\right) = \left\left(\sup A\right\right) + \left\left(\sup B\right\right).$ * If $A$ and $B$ are nonempty sets of positive real numbers then $\inf \left(A \cdot B\right) = \left\left(\inf A\right\right) \cdot \left\left(\inf B\right\right)$ and similarly for suprema $\sup \left(A \cdot B\right) = \left\left(\sup A\right\right) \cdot \left\left(\sup B\right\right).$ * If $S \subseteq \left(0, \infty\right)$ is non-empty and if $\frac := \left\,$ then $\frac = \inf_ \frac$ where this equation also holds when $\sup_ S = \infty$ if the definition $\frac := 0$ is used.The definition $\frac := 0$ is commonly used with the
extended real number In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra on ...
s; in fact, with this definition the equality $\frac = \inf_ \frac$ will also hold for any non-empty subset $S \subseteq \left(0, \infty\right].$ However, the notation $\frac$ is usually left undefined, which is why the equality $\frac = \sup_ \frac$ is given only for when $\inf_ S > 0.$
This equality may alternatively be written as $\frac = \inf_ \frac.$ Moreover, $\inf_ S = 0$ if and only if $\sup_ \frac = \infty,$ where if $\inf_ S > 0,$ then $\frac = \sup_ \frac.$

# Duality

If one denotes by $P^$ the partially-ordered set $P$ with the Converse relation, opposite order relation; that is, for all $x \text y,$ declare: $x \leq y \text P^ \quad \text \quad x \geq y \text P,$ then infimum of a subset $S$ in $P$ equals the supremum of $S$ in $P^$ and vice versa. For subsets of the real numbers, another kind of duality holds: $\inf S = - \sup \left(- S\right),$ where $-S := \.$

# Examples

## Infima

* The infimum of the set of numbers $\$ is $2.$ The number $1$ 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 In mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (math ...
of the set. * $\inf \ = 1.$ * $\inf \ = 0.$ * * $\inf \left\ = -1.$ * If $\left\left(x_n\right\right)_^$ is a decreasing sequence with limit $x,$ then $\inf x_n = x.$

## Suprema

* The supremum of the set of numbers $\$ is $3.$ The number $4$ is an upper bound, but it is not the least upper bound, and hence is not the supremum. * $\sup \ = \sup \ = 1.$ * $\sup \left\ = 1.$ * $\sup \ = \sup A + \sup B.$ * $\sup \left\ = \sqrt.$ In the last example, the supremum of a set of rationals is
irrational Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. Th ...
, which means that the rationals are
incomplete Incomplete may refer to: * Unfinished creative work * Gödel's incompleteness theorems, a specification of logic * Incomplete (Bad Religion song), "Incomplete" (Bad Religion song), 1994 * Incomplete (Sisqó song), "Incomplete" (Sisqó song), 1999 * ...
. One basic property of the supremum is $\sup \ ~\leq~ \sup \ + \sup \$ for any functionals $f$ and $g.$ The supremum of a subset $S$ of $\left(\N, \mid\,\right)$ where $\,\mid\,$ denotes "
divides In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

", is the lowest common multiple of the elements of $S.$ The supremum of a subset $S$ of $\left(P, \subseteq\right),$ where $P$ is the
power set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
of some set, is the supremum with respect to $\,\subseteq\,$ (subset) of a subset $S$ of $P$ is the union of the elements of $S.$