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Tuple
In mathematics, a tuple is a finite ordered list (sequence) of elements. An 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, or empty tuple, as it is referred to. An n-tuple is defined inductively using the construction of an ordered pair.
Mathematicians usually write tuples by listing the elements within parentheses "$({\text{ }})$" and separated by commas; 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 only used in defining arrays in some programming languages such as Java and Visual Basic, 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, 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" (three-fold) or "decuple" (ten-fold). This originates from medieval Latinplus (meaning "more") related to Greek -, which replaced the classical and late antique -plex (meaning "folded"), as in "duplex".^{[6]}^{[a]}
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.
A tuple may contain multiple instances of the same element, so tuple $(1,2,2,3)\neq (1,2,3)$; but set $\{1,2,2,3\}=\{1,2,3\}$.
Tuple elements are ordered: tuple $(1,2,3)\neq (3,2,1)$, but set $\{1,2,3\}=\{3,2,1\}$.
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
If we are dealing with sets, an n-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:
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; thus a 2-tuple
The 0-tuple (i.e. the empty tuple) is represented by the empty set $\emptyset$.
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):
The 0-tuple (i.e. the empty tuple) is represented by the empty set $\emptyset$;
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 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 m^{n}. 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 $S_{1},S_{2},\ldots ,S_{n}$ (note: the use of italics here that distinguishes sets from types) such that:
^Blackburn, Simon (2016) [1994]. "ordered n-tuple". The Oxford Dictionary of Philosophy. Oxford quick reference (3 ed.). Oxford: Oxford University Press. p. 342. ISBN9780198735304. Retrieved . ordered n-tuple[:] A generalization of the notion of an [...] ordered pair to sequences of n objects.
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