# Dictionary Definition

enumerate

### Verb

1 specify individually; "She enumerated the many
obstacles she had encountered"; "The doctor recited the list of
possible side effects of the drug" [syn: recite, itemize, itemise]

2 determine the number or amount of; "Can you
count the books on your shelf?"; "Count your change" [syn: count, number, numerate]

# User Contributed Dictionary

## English

### Verb

#### Translations

To specify each member of a sequence
individually in incrementing order

- Finnish: luetella

To determine the amount of

# 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 well-ordering.
Even if one does assume 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 enumerable is
often used to compare the relative sizes or cardinalities of different
sets. If one works in ZF (Zermelo-Fraenkel
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 well-ordered.
According to this characterization, an ordered enumeration is
defined to be a surjection 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.

## 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 non-injectivity 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

block out, book, calendar, call off, call over,
call the roll, catalog,
census, count, detail, enroll, enter, file, foliate, identify, impanel, index, inventory, itemize, keep score, list, measure, mention, number, numerate, outline, page, paginate, parse, particularize, pigeonhole, poll, post, program, quantify, quantize, recite, recount, register, relate, resolve, run over, scan, schedule, schematize, score, specialize, specify, tabulate, tale, tally, tell, tell off, tick
off