Tuple

For the musical term, see tuplet.

In mathematics, a tuple is a finite sequence of objects, that is, a list of a limited number of objects. Tuples are used to describe mathematical objects that consist of certain components. For example, a directed graph is defined as a tuple (V, E) where V is the set of nodes and E is a subset of V × V that denotes the edges.

The name "ordered list" is also used for design tuple.

Contents

1 Names of tuples
2 Names for tuples of specific length
3 Formal definitions
4 Usage in computer science
5 See also

Names of tuples

The term originated as an abstraction of the sequence: single, double, triple, quadruple, quintuple, n-tuple. A tuple of length n is usually described as an n-tuple. A 2-tuple is called a pair; a 3-tuple is a triple or triplet. The n can be any positive integer; thus one can for example say that a quaternion can be represented as a 4-tuple, and further constructed names are possible, such as octuple, but many mathematicians find it quicker to write "8-tuple", even if still pronouncing "octuple".

Names for tuples of specific length

Single (1) (or singleton or sole or only - many others)
Double (2) (or pair or twice)
Triple (3) (or triplet or treble or thrice or threesome or troika or trio)
Quadruple (4), quintuple or pentuple (5), sextuple or hextuple (6), septuple (7), octuple (8), nonuple (9) and decuple (10).

Formal definitions

The main properties that distinguish a tuple from, for example, a set are that (1) it can contain an object more than once and (2) the objects appear in a certain order. Note that (1) distinguishes it from an ordered set and that (2) distinguishes it from a multiset. This is often formalized by giving the following rule for the identity of two n-tuples:

(a₁, a₂, ...,a_n) = (b₁, b₂, ..., b_n) iff a₁ = b₁, a₂ = b₂ and so on.

Another way of formalizing tuples is by mapping them to more primitive constructs in set theory such as ordered pairs. For example, an n-tuple (with n > 2) can be defined as an ordered pair of its first entry and an (n−1)-tuple containing the remaining entries:

(a₁, a₂, ..., a_n) = (a₁, (a₂, ..., a_n))

Using the usual set-theoretic definition of an ordered pair and letting the empty set represent the empty tuple, this results in the following inductive definition:

the 0-tuple (i.e. the empty tuple) is represented by ∅
if x is an n-tuple then is an (n + 1)-tuple.

Using this definition, (1,2,2) would be

(1,(2,(2,()))) = (1,(2, )) = (1, }} ) = }}}}

There is an important similarity here with the way Lisp originally used the ordered pair abstraction to inductively create all of its n-tuple and list structures:

a special symbol NIL represents the empty list;
if X is a list and A an arbitrary value then the pair (A, X) represents a list with the head (i.e. first element) A and the tail (i.e. the remainder of the list without the head) X.

Usage in computer science

In computer science, tuple can have two distinct meanings. Typically in functional and some other programming languages, a tuple is a data object that holds several objects, similar to a mathematical tuple. Such an object is also known as a record.

In some languages, and especially in database theory, a tuple is usually defined as a finite function that maps field names to a certain value. Its purpose is the same as in mathematics, namely to indicate that a certain entity or object consists of certain components and/or has certain properties, but here these components are identified by a unique field name and not by a position, which often leads to a more user-friendly notation. The general term for this construct is an associative array; other programming languages have yet other names for the concept.

A small example of a tuple would be:

( player : "Harry", score : 25 )

which is a function that maps the field name "player" to the string "Harry" and the field name "score" to the number 25. Note that the order of the components is not relevant, so the same tuple can also be written as:

( score : 25, player : "Harry" )

In the relational model such tuples are typically used to represent a single proposition, in this case that there exists a player with the name "Harry" and a score of 25.

In programming, languages tuples are used to form data structures. For example, the following could be a structure that represents a node in a doubly linked list:

( value : 16, previous-node : 1174782, next-node : 1174791 )

Tuple

Names of tuples

Names for tuples of specific length

Formal definitions

Usage in computer science

See also