For the musical term, see
Tuplet."Octuple" redirects here. For the boat, see
Octuple scull."Duodecuple" redirects here. For the musical technique, see
Twelve-tone technique."Sextuple" redirects here. For the sporting achievement of association football, see
Sextuple (association football).
In
mathematics, a tuple is a finite
sequence or ordered list of
numbers or, more generally,
mathematical objects, which are called the elements of the tuple. An n-tuple is a tuple of n elements, where n is a non-negative
integer. There is only one 0-tuple, called the empty tuple. A 1-tuple and a 2-tuple are commonly called a
singleton and an
ordered pair, respectively.
Tuple may be formally defined from ordered pairs by
recurrence by starting from
ordered pairs; indeed, an n-tuple can be identified with the ordered pair of its (n − 1) first elements and its nth element.
Tuples are usually written by listing the elements within parentheses "( )", separated by a comma and a space; for example, (2, 7, 4, 1, 7) denotes a 5-tuple. Sometimes other symbols are used to surround the elements, such as square brackets "[ ]" or angle brackets "⟨ ⟩". Braces "{ }" are used to specify
arrays in some programming languages but not in mathematical expressions, as they are the standard notation for
sets. The term tuple can often occur when discussing other mathematical objects, such as
vectors.
The term originated as an abstraction of the sequence: single, couple/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 or empty tuple. A 1‑tuple is called a single (or singleton), a 2‑tuple is called an ordered pair or couple, and a 3‑tuple is called a triple (or triplet). The number n can be any nonnegative
integer. For example, a
complex number can be represented as a 2‑tuple of reals, 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 ‑uple as the suffix, the original suffix was ‑ple as in "triple" (three-fold) or "decuple" (ten‑fold). This originates from
medieval Latinplus (meaning "more") related to
Greek ‑πλοῦς, which replaced the classical and late antique ‑plex (meaning "folded"), as in "duplex".[6][a]
Properties
The general rule for the identity of two n-tuples is
Functions are commonly identified with their
graphs, which is a certain set of ordered pairs.
Indeed, many authors use graphs as the definition of a function.
Using this definition of "function", the above function can be defined as:
Tuples as nested ordered pairs
Another way of modeling tuples in Set Theory is as nested
ordered pairs. This approach assumes that the notion of ordered pair has already been defined.
The 0-tuple (i.e. the empty tuple) is represented by the empty set .
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):
This definition can be applied recursively to the (n − 1)-tuple:
Thus, for example:
A variant of this definition starts "peeling off" elements from the other end:
The 0-tuple (i.e. the empty tuple) is represented by the empty set ;
Let be an n-tuple , and let . Then, . (The right arrow, , could be read as "adjoined with".)
In this formulation:
n-tuples of m-sets
In
discrete mathematics, especially
combinatorics and finite
probability theory, n-tuples arise in the context of various counting problems and are treated more informally as ordered lists of length n.[7]n-tuples whose entries come from a set of m elements are also called arrangements with repetition, permutations of a multiset and, in some non-English literature, variations with repetition. The number of n-tuples of an m-set is mn. This follows from the combinatorial
rule of product.[8] If S is a finite set of
cardinalitym, this number is the cardinality of the n-fold
Cartesian powerS × S × ⋯ × S. Tuples are elements of this product set.
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 (note: the use of italics here that distinguishes sets from types) such that:
and the interpretation of the basic terms is:
.
The n-tuple of type theory has the natural interpretation as an n-tuple of set theory:[10]
The
unit type has as semantic interpretation the 0-tuple.
^Blackburn, Simon (1994). "ordered n-tuple".
The Oxford Dictionary of Philosophy. Oxford guidelines quick reference (3 ed.). Oxford: Oxford University Press (published 2016). p. 342.
ISBN9780198735304. Retrieved 2017-06-30. ordered n-tuple[:] A generalization of the notion of an [...] ordered pair to sequences of n objects.