Tuple

This article is about the mathematical concept. For the musical term, see Tuplet. For the achievement in Association Football, see The Sextuple.

In mathematics and computer science, 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 positive integer. There is also 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 "( )" and separated by commas; for example, (2,7,4,1,7) denotes a 5-tuple. Sometimes other delimiters are used, such as square brackets "[ ]" or angle brackets "$\langle\text{ }\rangle$". 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 algebra, a ring is commonly defined as a 3-tuple $(E,+,\times)$, where E is some set, and " + ", and "$\times$" are functions mapping the Cartesian product $E \times E$ to E with specific properties. 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.

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 a pair and a 3‑tuple is a triple or triplet. The 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 octuple, (many mathematicians write the abbreviation "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" (three-fold) or "decuple" (ten‑fold). This originates from a medieval Latin suffix ‑plus (meaning "more") related to Greek ‑πλοῦς, which replaced the classical and late antique ‑plex (meaning "folded").^[1]

Formal definitions

Characteristic properties

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

$(a_1, a_2, \ldots, a_n) = (b_1, b_2, \ldots, b_n)$ if and only if $a_1=b_1,\text{ }a_2=b_2,\text{ }\ldots,\text{ }a_n=b_n$

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

1. A tuple may contain multiple instances of the same element, so
   tuple $(1,2,2,3) \neq (1,2,3)$; but multiset {1,2,2,3} = set {1,2,3}.
2. Tuple elements are ordered: tuple $(1,2,3) \neq (3,2,1)$, but set {1,2,3} = {3,2,1}.
3. A tuple has a finite number of elements, while a set or a multiset may have an infinite number of elements.

Tuples as functions

An n-tuple can also 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:

$(a_1, a_2, \dots, a_n) \equiv (X,Y,F)$

where:

$\begin{align} X & = \{1, 2, \dots, n\} \\ Y & = \{a_1, a_2, \ldots, a_n\} \\ F & = \{(1, a_1), (2, a_2), \ldots, (n, a_n)\} \\ \end{align}$

Tuples as nested ordered pairs

Another way of formalizing tuples is as nested ordered pairs:

1. The 0-tuple (i.e. the empty tuple) is represented by the empty set $\emptyset$.
2. An n-tuple, with n > 0, can be defined as an ordered pair of its first entry and an (n − 1)-tuple (which contains the remaining entries when n > 1):

   $(a_1, a_2, a_3, \ldots, a_n) = (a_1, (a_2, a_3, \ldots, a_n))$

This definition can be applied recursively to the (n − 1)-tuple:

$(a_1, a_2, a_3, \ldots, a_n) = (a_1, (a_2, (a_3, (\ldots, (a_n, \emptyset)\ldots))))$

Thus, for example:

$\begin{align} (1, 2, 3) & = (1, (2, (3, \emptyset))) \\ (1, 2, 3, 4) & = (1, (2, (3, (4, \emptyset)))) \\ \end{align}$

A variant of this definition starts "peeling off" elements from the other end:

1. The 0-tuple is the empty set $\emptyset$.
2. For n > 0:

   $(a_1, a_2, a_3, \ldots, a_n) = ((a_1, a_2, a_3, \ldots, a_{n-1}), a_n)$

This definition can be applied recursively:

$(a_1, a_2, a_3, \ldots, a_n) = ((\ldots(((\emptyset, a_1), a_2), a_3), \ldots), a_n)$

Thus, for example:

$\begin{align} (1, 2, 3) & = (((\emptyset, 1), 2), 3) \\ (1, 2, 3, 4) & = ((((\emptyset, 1), 2), 3), 4) \\ \end{align}$

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:

1. The 0-tuple (i.e. the empty tuple) is represented by the empty set $\emptyset$;
2. Let x be an n-tuple $(a_1, a_2, \ldots, a_n)$, and let $x \rightarrow b \equiv (a_1, a_2, \ldots, a_n, b)$. Then, $x \rightarrow b \equiv \{\{x\}, \{x, b\}\}$. (The right arrow, $\rightarrow$, could be read as "adjoined with".)

In this formulation:

$\begin{array}{lclcl} () & & &=& \emptyset \\ & & & & \\ (1) &=& () \rightarrow 1 &=& \{\{()\},\{(),1\}\} \\ & & &=& \{\{\emptyset\},\{\emptyset,1\}\} \\ & & & & \\ (1,2) &=& (1) \rightarrow 2 &=& \{\{(1)\},\{(1),2\}\} \\ & & &=& \{\{\{\{\emptyset\},\{\emptyset,1\}\}\}, \\ & & & & \{\{\{\emptyset\},\{\emptyset,1\}\},2\}\} \\ & & & & \\ (1,2,3) &=& (1,2) \rightarrow 3 &=& \{\{(1,2)\},\{(1,2),3\}\} \\ & & &=& \{\{\{\{\{\{\emptyset\},\{\emptyset,1\}\}\}, \\ & & & & \{\{\{\emptyset\},\{\emptyset,1\}\},2\}\}\}, \\ & & & & \{\{\{\{\{\emptyset\},\{\emptyset,1\}\}\}, \\ & & & & \{\{\{\emptyset\},\{\emptyset,1\}\},2\}\},3\}\} \\ \end{array}$

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,^[2] 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}.^[2]

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.^[2] 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

Main article: Product type

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:

$(x_1, x_2, \ldots, x_n) : \text{T}_1 \times \text{T}_2 \times \ldots \times \text{T}_n$

and the projections are term constructors:

$\pi_1(x) : \text{T}_1,\text{ }\pi_2(x) : \text{T}_2,\text{ }\ldots,\text{ }\pi_n(x) : \text{T}_n$

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.^[3]

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 $S_1, S_2, \ldots, S_n$ (note: the use of italics here that distinguishes sets from types) such that:

$[\![\text{T}_1]\!] = S_1,\text{ }[\![\text{T}_2]\!] = S_2,\text{ }\ldots,\text{ }[\![\text{T}_n]\!] = S_n$

and the interpretation of the basic terms is:

$[\![x_1]\!] \in [\![\text{T}_1]\!],\text{ }[\![x_2]\!] \in [\![\text{T}_2]\!],\text{ }\ldots,\text{ }[\![x_n]\!] \in [\![\text{T}_n]\!]$.

The n-tuple of type theory has the natural interpretation as an n-tuple of set theory:^[4]

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

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

See also

• Arity
• Exponential object
• Formal language
• OLAP: Multidimensional Expressions
• Relation (mathematics)
• Tuplespace

Notes

1. ^ OED, s.v. "triple", "quadruple", "quintuple", "decuple"
2. ^ ^{a b c} Serge Abiteboul, Richard Hull, Victor Vianu, Foundations of databases, Addison-Wesley, 1995, ISBN 0201537710, p. 29–33
3. ^ Pierce, Benjamin (2002). Types and Programming Languages. MIT Press. pp. 126–132. ISBN 0-262-16209-1. 
4. ^ Steve Awodey, From sets, to types, to categories, to sets, 2009, preprint

References

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

• Abraham Adolf Fraenkel, Yehoshua Bar-Hillel, Azriel Lévy, Foundations of set theory, Elsevier Studies in Logic Vol. 67, Edition 2, revised, 1973, ISBN 0720422701, p. 33
• Gaisi Takeuti, W. M. Zaring, Introduction to Axiomatic Set Theory, Springer GTM 1, 1971, ISBN 978-0-387-90024-7, p. 14
• George J. Tourlakis, Lecture Notes in Logic and Set Theory. Volume 2: Set theory, Cambridge University Press, 2003, ISBN 978-0-521-75374-6, pp. 182-193
• Keith Devlin, The Joy of Sets. Springer Verlag, 2nd ed., 1993, ISBN 0-387-94094-4, pp. 7-8

External links

• Prefixes & Words Based On Latin Number Names