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Tuple

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A finite sequence (admitting repetitions) of elements from some set . A tuple is denoted by \langle x_1,\ldots,x_n\rangle, (x_i), (x_i)_{i=1}^n, (x_i)_1^n, (x_i)_{i\in\{1,\ldots,n\}}, (x_1,\ldots,x_n), or x_1,\ldots,x_n. The number n is called its length (n\geq0), x_i is called the i-th term of the n-tuple and x_i\in X (1\leq i\leq n). For n=0 one finds the empty tuple, which contains no terms.

Synonyms of the term tuple are the following: a word in the alphabet X (in this case it is usually assumed that X is finite); an element of some Cartesian power of the set X; an element of the free monoid (free semi-group with identity) generated by X; a function defined on the first n natural numbers (n\geq0) with values in X.


Comments

The typical property of tuples is that a tuple (x_1,\ldots,x_n) is equal to another one (y_1,\ldots,y_m) precisely when n=m and x_i=y_i for all i=1,\ldots,n. In the context of a set-theoretical foundation of mathematics (such as Zermelo–Fraenkel set theory), where every object must be a set or a class, tuples are usually constructed as sets by the following inductive procedure: (x_1,\ldots,x_n) is the empty set for n=0, and (x_1,\ldots,x_{n+1})=\{(x_1,\ldots,x_n),\{x_{n+1}\}\}.

See also: Multiset

How to Cite This Entry:
Tuple. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tuple&oldid=37507
This article was adapted from an original article by V.N. Grishin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article