A tuple is an ordered list of elements.
In set theory, an [ordered] n-tuple is a sequence of elements, where is a non-negative integer.
There is only one 0-tuple, an empty sequence.
An -tuple is defined inductively using the construction of an ordered pair.
Tuples are usually written by listing the elements within parentheses "" and separated
by commas; for example, denotes a 5-tuple.
Sometimes other delimiters are used, such as square brackets "" or angle brackets "". Braces
"" are almost never used for tuples, as they are the standard notation for sets.
Tuples are often used to describe other mathematical objects, such as vectors.
In computer science, tuples are directly implemented as product types in most functional programming
languages.
More commonly, they are implemented as record types, where the components are labeled instead
of being identified by position alone.
This approach is also used in relational algebra.
Tuples are also used in relation to programming the semantic web with Resource Description
Framework or RDF.
Tuples are also used in linguistics and philosophy.
Etymology The term originated as an abstraction of the
sequence: single, double, triple, quadruple, quintuple, sextuple, septuple, octuple, ..., n‑tuple,
..., where the prefixes are taken from the Latin names of the numerals.
The unique 0‑tuple is called the null tuple.
A 1‑tuple is called a singleton, a 2‑tuple is called an ordered pair and a 3‑tuple
is a triple or triplet.
n can be any nonnegative integer.
For example, a complex number can be represented as a 2‑tuple, a quaternion can be represented
as a 4‑tuple, an octonion can be represented as an 8‑tuple and a sedenion can be represented
as a 16‑tuple.
Although these uses treat ‑tuple as the suffix, the original suffix was ‑ple as
in "triple" or "decuple".
This originates from a medieval Latin suffix ‑plus related to Greek ‑πλοῦς, which
replaced the classical and late antique ‑plex, as in "duplex".
Names for tuples of specific lengths Properties
The general rule for the identity of two -tuples is
if and only if Thus a tuple has properties that distinguish
it from a set.
A tuple may contain multiple instances of the same element, so
tuple ; but set . Tuple elements are ordered: tuple , but set
. A tuple has a finite number of elements, while
a set or a multiset may have an infinite number of elements.
Definitions There are several definitions of tuples that
give them the properties described in the previous section.
Tuples as functions An -tuple can be regarded as a function, F,
whose domain is the tuple's implicit set of element indices, X, and whose codomain, Y,
is the tuple's set of elements.
Formally:
where:
Tuples as nested ordered pairs Another way of formalizing tuples is as nested
ordered pairs.
This approach assumes that the notion of ordered pair has already been defined; thus a 2-tuple
The 0-tuple is represented by the empty set .
An -tuple, with , can be defined as an ordered pair of its first entry and an -tuple:
This definition can be applied recursively to the -tuple:
Thus, for example:
A variant of this definition starts "peeling off" elements from the other end:
The 0-tuple is the empty set . For :
This definition can be applied recursively:
Thus, for example:
Tuples as nested sets Using Kuratowski's representation for an ordered
pair, the second definition above can be reformulated in terms of pure set theory:
The 0-tuple is represented by the empty set ;
Let be an -tuple , and let . Then, . In this formulation:
Relational model In database theory, the relational model uses
a tuple definition similar to tuples as functions, but each tuple element is identified by a
distinct name, called an attribute, instead of a number; this leads to a more user-friendly
and practical notation.
A tuple in the relational model is formally defined as a finite function that maps attributes
to values.
For example: (player : "Harry", score : 25)
In this notation, attribute–value pairs may appear in any order.
The distinction between tuples in the relational model and those in set theory is only superficial;
the above example can be interpreted as a 2-tuple if an arbitrary total order is imposed
on the attributes and then the elements are distinguished by this ordering rather than
by the attributes themselves.
Conversely, a 2-tuple may be interpreted as relational model tuple over the attributes
. In the relational model, a relation is a finite
set of tuples all having the same finite set of attributes.
This set of attributes is more formally called the sort of the relation, or more casually
referred to as the set of column names.
A tuple is usually implemented as a row in a database table, but see relational algebra
for means of deriving tuples not physically represented in a table.
Type theory
In type theory, commonly used in programming languages, a tuple has a product type; this
fixes not only the length, but also the underlying types of each component.
Formally:
and the projections are term constructors:
The tuple with labeled elements used in the relational model has a record type.
Both of these types can be defined as simple extensions of the simply typed lambda calculus.
The notion of a tuple in type theory and that in set theory are related in the following
way: If we consider the natural model of a type theory, and use the Scott brackets to
indicate the semantic interpretation, then the model consists of some sets such that:
and the interpretation of the basic terms is:
. The -tuple of type theory has the natural
interpretation as an -tuple of set theory:
The unit type has as semantic interpretation the 0-tuple.
See also Arity
Exponential object Formal language
OLAP: Multidimensional Expressions Prime k-tuple
Relation Tuplespace
Notes
References The set theory definitions herein are found
in any textbook on the topic, e.g.