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An enumeration is a complete, ordered listing of all the items in a collection. The term is commonly used 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 ...
and
computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of , , and . Computer science ...
to refer to a listing of all of the
element Element may refer to: Science * Chemical element Image:Simple Periodic Table Chart-blocks.svg, 400px, Periodic table, The periodic table of the chemical elements In chemistry, an element is a pure substance consisting only of atoms that all ...
s of a set. The precise requirements for an enumeration (for example, whether the set must be
finite Finite is the opposite of Infinity, infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected ...
, or whether the list is allowed to contain repetitions) depend on the discipline of study and the context of a given problem. Some sets can be enumerated by means of a natural ordering (such as 1, 2, 3, 4, ... for the set of
positive integer 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). ...
s), but in other cases it may be necessary to impose a (perhaps arbitrary) ordering. In some contexts, such as enumerative combinatorics, the term ''enumeration'' is used more in the sense of ''
counting Counting is the process of determining the number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) ...
'' – with emphasis on determination of the number of elements that a set contains, rather than the production of an explicit listing of those elements.


Combinatorics

In combinatorics, enumeration means
counting Counting is the process of determining the number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) ...
, i.e., determining the exact number of elements of finite sets, usually grouped into infinite families, such as the family of sets each consisting of all
permutation In , a permutation of a is, loosely speaking, an arrangement of its members into a or , or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order o ...

permutation
s of some finite set. There are flourishing subareas in many branches of mathematics concerned with enumerating in this sense objects of special kinds. For instance, in '' partition enumeration'' and ''
graph enumeration In combinatorics, an area of 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 (mathematica ...
'' the objective is to count partitions or graphs that meet certain conditions.


Set theory

In
set theory Set theory is the branch of that studies , which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of , is mostly concerned with those that are relevant to ...
, the notion of enumeration has a broader sense, and does not require the set being enumerated to be finite.


Listing

When an enumeration is used in an
ordered list
ordered list
context, we impose some sort of ordering structure requirement on the
index set In mathematics, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a Set (mathematics), set may be ''indexed'' or ''labeled'' by means of the elements of a set , then is an index set. Th ...
. While we can make the requirements on the ordering quite lax in order to allow for great generality, the most natural and common prerequisite is that the index set be
well-ordered 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 ...
. According to this characterization, an ordered enumeration is defined to be a surjection (an onto relationship) with a well-ordered domain. This definition is natural in the sense that a given well-ordering on the index set provides a unique way to list the next element given a partial enumeration.


Countable vs. uncountable

The most common use of enumeration in set theory occurs in the context where infinite sets are separated into those that are countable and those that are not. In this case, an enumeration is merely an enumeration with domain ''ω'', the ordinal of the
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
s. This definition can also be stated as follows: * As a
surjective 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 ...
mapping from \mathbb (the
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
s) to ''S'' (i.e., every element of ''S'' is the image of at least one natural number). This definition is especially suitable to questions of
computability Computability is the ability to solve a problem in an effective manner. It is a key topic of the field of computability theory Computability theory, also known as recursion theory, is a branch of mathematical logic Mathematical logic, also calle ...
and elementary
set theory Set theory is the branch of that studies , which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of , is mostly concerned with those that are relevant to ...
. We may also define it differently when working with finite sets. In this case an enumeration may be defined as follows: * As a
bijective 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 ...
mapping from ''S'' to an initial segment of the natural numbers. This definition is especially suitable to combinatorial questions and finite sets; then the initial segment is for some ''n'' which is the
cardinality 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 ...
of ''S''. In the first definition it varies whether the mapping is also required to be
injective 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 ...
(i.e., every element of ''S'' is the image of ''exactly one'' natural number), and/or allowed to be
partial Partial may refer to: Mathematics *Partial derivative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...

partial
(i.e., the mapping is defined only for some natural numbers). In some applications (especially those concerned with computability of the set ''S''), these differences are of little importance, because one is concerned only with the mere existence of some enumeration, and an enumeration according to a liberal definition will generally imply that enumerations satisfying stricter requirements also exist. Enumeration of
finite set 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 ...
s obviously requires that either non-injectivity or partiality is accepted, and in contexts where finite sets may appear one or both of these are inevitably present.


Examples


Properties

* There exists an enumeration for a set (in this sense) if and only if the set is
countable 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 ...
. * If a set is enumerable it will have an
uncountable 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 ...

uncountable
infinity of different enumerations, except in the degenerate cases of the empty set or (depending on the precise definition) sets with one element. However, if one requires enumerations to be injective ''and'' allows only a limited form of partiality such that if ''f''(''n'') is defined then ''f''(''m'') must be defined for all ''m'' < ''n'', then a finite set of ''N'' elements has exactly ''N''! enumerations. * An enumeration ''e'' of a set ''S'' with domain \mathbb induces a
well-order In mathematics, a well-order (or well-ordering or well-order relation) on a Set (mathematics), set ''S'' is a total order on ''S'' with the property that every non-empty subset of ''S'' has a least element in this ordering. The set ''S'' together ...
≤ on that set defined by ''s'' ≤ ''t'' if and only if \min e^(s) \leq \min e^(t). Although the order may have little to do with the underlying set, it is useful when some order of the set is necessary.


Ordinals

In
set theory Set theory is the branch of that studies , which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of , is mostly concerned with those that are relevant to ...
, there is a more general notion of an enumeration than the characterization requiring the domain of the listing function to be an
initial segment A Hasse diagram of the power set of the set with the upper set ↑ colored green. The white sets form the lower set ↓. In mathematics, an upper set (also called an upward closed set, an upset, or an isotone set in ''X'') of a partially ordered s ...
of the Natural numbers where the domain of the enumerating function can assume any
ordinal Ordinal may refer to: * Ordinal data, a statistical data type consisting of numerical scores that exist on an arbitrary numerical scale * Ordinal date, a simple form of expressing a date using only the year and the day number within that year * Or ...
. Under this definition, an enumeration of a set ''S'' is any
surjection 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 ...

surjection
from an ordinal α onto ''S''. The more restrictive version of enumeration mentioned before is the special case where α is a finite ordinal or the first limit ordinal ω. This more generalized version extends the aforementioned definition to encompass transfinite listings. Under this definition, the first uncountable ordinal \omega_1 can be enumerated by the identity function on \omega_1 so that these two notions do not coincide. More generally, it is a theorem of ZF that any
well-ordered 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 ...
set can be enumerated under this characterization so that it coincides up to relabeling with the generalized listing enumeration. If one also assumes the
Axiom of Choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

Axiom of Choice
, then all sets can be enumerated so that it coincides up to relabeling with the most general form of enumerations. Since set theorists work with infinite sets of arbitrarily large
cardinalities In mathematics, the cardinality of a set (mathematics), set is a measure of the "number of Element (mathematics), elements" of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 1 ...
, the default definition among this group of mathematicians of an enumeration of a set tends to be any arbitrary α-sequence exactly listing all of its elements. Indeed, in Jech's book, which is a common reference for set theorists, an enumeration is defined to be exactly this. Therefore, in order to avoid ambiguity, one may use the term finitely enumerable or denumerable to denote one of the corresponding types of distinguished countable enumerations.


Comparison of cardinalities

Formally, the most inclusive definition of an enumeration of a set ''S'' is any
surjection 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 ...

surjection
from an arbitrary
index set In mathematics, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a Set (mathematics), set may be ''indexed'' or ''labeled'' by means of the elements of a set , then is an index set. Th ...
''I'' onto ''S''. In this broad context, every set ''S'' can be trivially enumerated by the
identity function Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function (mathematics), function that always returns the same value that was ...

identity function
from ''S'' onto itself. If one does ''not'' assume the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

axiom of choice
or one of its variants, ''S'' need not have any
well-ordering 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 ...
. Even if one does assume the axiom of choice, ''S'' need not have any natural well-ordering. This general definition therefore lends itself to a counting notion where we are interested in "how many" rather than "in what order." In practice, this broad meaning of enumeration is often used to compare the relative sizes or
cardinalities In mathematics, the cardinality of a set (mathematics), set is a measure of the "number of Element (mathematics), elements" of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 1 ...
of different sets. If one works in
Zermelo–Fraenkel set theory In set theory illustrating the intersection of two sets Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set ...
without the axiom of choice, one may want to impose the additional restriction that an enumeration must also be
injective 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 ...
(without repetition) since in this theory, the existence of a surjection from ''I'' onto ''S'' need not imply the existence of an
injection Injection or injected may refer to: Science and technology * Injection (medicine) An injection (often referred to as a "shot" in US English, a "jab" in UK English, or a "jag" in Scottish English and Scots Language, Scots) is the act of adminis ...
from ''S'' into ''I''.


Computability and complexity theory

In
computability theory Computability theory, also known as recursion theory, is a branch of mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. ...
one often considers countable enumerations with the added requirement that the mapping from \mathbb (set of all natural numbers) to the enumerated set must be
computable Computability is the ability to solve a problem in an effective manner. It is a key topic of the field of computability theory Computability theory, also known as recursion theory, is a branch of mathematical logic Mathematical logic, also calle ...
. The set being enumerated is then called
recursively enumerable In computability theory, traditionally called recursion theory, a set ''S'' of natural numbers is called recursively enumerable, computably enumerable, semidecidable, partially decidable, listable, provable or Turing-recognizable if: *There is a ...
(or computably enumerable in more contemporary language), referring to the use of
recursion theory Computability theory, also known as recursion theory, is a branch of mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. ...
in formalizations of what it means for the map to be computable. In this sense, a subset of the natural numbers is computably enumerable if it is the range of a computable function. In this context, enumerable may be used to mean computably enumerable. However, these definitions characterize distinct classes since there are uncountably many subsets of the natural numbers that can be enumerated by an arbitrary function with domain ω and only countably many computable functions. A specific example of a set with an enumeration but not a computable enumeration is the complement of the halting set. Furthermore, this characterization illustrates a place where the ordering of the listing is important. There exists a computable enumeration of the halting set, but not one that lists the elements in an increasing ordering. If there were one, then the halting set would be decidable, which is provably false. In general, being recursively enumerable is a weaker condition than being a
decidable set The word ''decidable'' may refer to: * Decidable language *Decidability (logic) for the equivalent in mathematical logic *Decidable problem and Undecidable problem *Gödel's incompleteness theorem, a theorem on the indecidability of languages cons ...
. The notion of enumeration has also been studied from the point of view of
computational complexity theory Computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved by a computer. A computation problem is solvable by ...
for various tasks in the context of enumeration algorithms.


See also

*
Ordinal number In set theory Set theory is the branch of that studies , which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of , is mostly concerned with those that ...
*
Enumerative definition An enumerative definition of a concept or term is a special type of extensional definition that gives an explicit and exhaustive listing of all the objects that fall under the concept or term in question. Enumerative definitions are only possible ...
*
Sequence 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 t ...

Sequence


References

*


External links

* {{Authority control Enumerative combinatorics Mathematical logic Ordering