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In mathematics, computer science, linguistics,[1] and philosophy[2] a tuple is an ordered list of elements. In set theory, an (ordered) n-tuple is a sequence (or ordered list) of n elements, where n is a non-negative integer. There is only one 0-tuple, an empty sequence. An n-tuple is defined inductively using the construction of an ordered pair. Tuples are usually written by listing the elements within parentheses "(\text{ })" and separated by commas; for example, (2, 7, 4, 1, 7) denotes a 5-tuple. Sometimes other delimiters are used, such as square brackets "[\text{ }]" or angle brackets "\langle\text{ }\rangle". Braces "\{\}" are almost never used for tuples, as they are the standard notation for sets.





Names for tuples of specific lengths[edit]

Tuple Length n Name Alternative names
0 empty tuple unit
1 single Monad
2 double couple / pair / dual / twin
3 triple treble
4 quadruple quad
5 quintuple pentuple
6 sextuple hextuple
7 septuple
8 octuple
9 nonuple
10 decuple
11 undecuple hendecuple
12 duodecuple
13 tredecuple
100 centuple


The general rule for the identity of two n-tuples is


Thus a tuple has properties that distinguish it from a set.




Tuples as functions[edit]





Tuples as nested ordered pairs[edit]

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




Thus, for example:



  2. For n > 0:

This definition can be applied recursively:


Thus, for example:


Tuples as nested sets[edit]



In this formulation:

     ()      & &                     &=& \emptyset                                    \\
             & &                     & &                                              \\
             & &                     & &                                              \\
             & &                     & &                                              \\

Relational model[edit]

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.[4] 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 (e.g. player \leq score) 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 \{1, 2\}.[4]

In the relational model, a relation is a (possibly empty) 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.[4] 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[edit]

Main article: Product type



and the projections are term constructors:





and the interpretation of the basic terms is:



[\![(x_1, x_2, \ldots, x_n)]\!] = ([\![x_1]\!], [\![x_2]\!], \ldots, [\![x_n]\!])

The unit type has as semantic interpretation the 0-tuple.

See also[edit]


  1. ^ http://www.oxfordreference.com/view/10.1093/acref/9780199202720.001.0001/acref-9780199202720-e-2276
  2. ^ http://www.oxfordreference.com/view/10.1093/acref/9780199541430.001.0001/acref-9780199541430-e-2262
  4. ^ a b c Serge Abiteboul, Richard Hull, Victor Vianu, Foundations of databases, Addison-Wesley, 1995, ISBN 0-201-53771-0, p. 29–33


The set theory definitions herein are found in any textbook on the topic, e.g.