# Tuple

A finite sequence (admitting repetitions) of elements from some set . A tuple is denoted by , , , , , , or . The number is called its length (), is called the -th term of the -tuple and (). For one finds the empty tuple, which contains no terms.

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

#### Comments

The typical property of tuples is that a tuple is equal to another one precisely when and for all . 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: is the empty set for , and .

**How to Cite This Entry:**

Tuple.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Tuple&oldid=11398