Dictionary Definition
enumeration
Noun
1 a numbered list [syn: numbering]
2 the act of counting; "the counting continued
for several hours" [syn: count, counting, numeration, reckoning, tally]
User Contributed Dictionary
English
Pronunciation

 Rhymes: eɪʃǝn
Noun
 The act of enumerating, making separate mention, or recounting.
 A detailed account, in which each thing is specially noticed.
 A recapitulation, in the peroration, of the heads of an argument.
 For example  Monday, Tuesday, Wednesday is an enumeration of days.
Anagrams
Extensive Definition
In mathematics and theoretical
computer
science, the broadest and most abstract definition of an
enumeration of a set is an exact listing of all of its elements
(perhaps with repetition). The restrictions imposed on the type of
list used depend on the branch of mathematics and the context in
which one is working. In more specific settings, this notion of
enumeration encompasses the two different types of listing: one
where there is a natural ordering and one where the ordering is
more nebulous. These two different kinds of enumerations correspond
to a procedure for listing all members of the set in some definite
sequence, or a count of objects of a specified kind, respectively.
While the two kinds of enumeration often overlap in most natural
situations, they can assume very different meanings in certain
contexts.
Enumeration as counting
Formally, the most inclusive definition of an
enumeration of a set S is any surjection from an arbitrary
index set I onto S. In this broad context, every set S can be
trivially enumerated by the identity function from S onto itself.
If one does not assume the Axiom of
Choice or one of its variants, S need not have any wellordering.
Even if one does assume Choice, S need not have any natural
wellordering.
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 enumerable is
often used to compare the relative sizes or cardinalities of different
sets. If one works in ZF (ZermeloFraenkel
set theory without choice), one may want to impose the
additional restriction that an enumeration must also be injective (without repetition)
since in this theory, the existence of a surjection from I onto S
need not imply the existence of an injection from S into I.
Enumeration as listing
When an enumeration is used in an ordered list
context, we impose some sort of ordering structure requirement on
the index set. 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 wellordered.
According to this characterization, an ordered enumeration is
defined to be a surjection with a wellordered domain. This
definition is natural in the sense that a given wellordering on
the index set provides a unique way to list the next element given
a partial enumeration.
Enumeration in countable vs. uncountable context
The most common use of enumeration 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. This definition can
also be stated as follows:
 As a surjective mapping from \mathbb (the natural numbers) 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 and elementary set theory.
We may also define it differently when working
with finite sets. In this case an enumeration may be defined as
follows:
 As a bijective 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 of S.
In the first definition it varies whether the
mapping is also required to be injective (i.e., every element
of S is the image of exactly one natural number), and/or allowed to
be 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 sets obviously requires
that either noninjectivity or partiality is accepted, and in
contexts where finite sets may appear one or both of these are
inevitably present.
Examples
 The natural numbers are enumerable by the function f(x) = x. In this case f: \mathbb \to \mathbb is simply the identity function.
 \mathbb, the set of integers is enumerable by
 f(x):= \begin (x+1)/2, & \mbox x \mbox \\ x/2, & \mbox x \mbox. \end
f: \mathbb \to \mathbb is a bijection since every
natural number corresponds to exactly one integer. The following
table gives the first few values of this enumeration:
Synonyms, Antonyms and Related Words
accounting, active list,
blacklist, blocking, census, checklist, civil list,
counting, dactylonomy, detailing, foliation, inventory, inventorying, itemization, items, list, measurement, numbering, numeration, pagination, parsing, quantification, quantization, register, registry, repertory, resolution, retired list,
scansion, schedule, schematization, sick
list, tally, tally sheet,
tallying, telling