Tuple

This article is about the mathematical term. For the term tuple in the field of databases, see Tuple (computer science).

Tuples (derived from the Middle Latin quintuplus 'fivefold', septuplus 'sevenfold', centuplus 'hundredfold', etc.) are, along with sets, an important way of grouping mathematical objects in mathematics. A tuple consists of a list of finitely many objects, not necessarily different from each other. Here, unlike sets, the order of the objects matters. There are several ways to formally represent tuples as sets. Tuples are used in many areas of mathematics, for example as coordinates of points or as vectors in multidimensional vector spaces.

Tuples independent of their length are rarely mentioned. Rather, one uses the word n -tuples and the special cases of them mentioned in the next section when the length nresults from the context as a fixed number or as a named constant like If, on the other hand, one considers many finite sequences of different lengths of elements of a basic set, one speaks of finite sequences or defines a new term often compounded with "chain", e.g. string, addition chain.

In computer science, the term tuple is also used as a synonym for a data set. In various programming languages, such as Python, tuples are immutable data sets.

Notation

An n-tuple is a collection of nmathematical objects x_{1},\ldots ,x_{n} in a list. Unlike sets, the objects do not necessarily have to be different from each other and their order is important. Tuples are usually represented by round brackets.

(x_{1},\ldots ,x_{n})

where two consecutive objects are separated by a comma. The object at the i ix_{i}position is called the i-th component of the tuple. Occasionally, other types of brackets, such as angle brackets or angle brackets, are used for notation:

{\displaystyle \langle x_{1},\ldots ,x_{n}\rangle }or {\displaystyle [x_{1},\ldots ,x_{n}]}

Other separators, such as semicolons or vertical bars, are also common. Other notation variants are

(x_{i})_{{i=1,\ldots ,n}},(x_{i})_{{i\in \{1,\ldots ,n\}}},(x_{i})_{{i=1}}^{n}

or for short (x_{i})if the length of the tuple is clear from the context.

Special designations for n-tuples with small n

A 2-tuple is also called an ordered pair or duple, a 3-tuple also a triple, a 4-tuple also a quadruple, a 5-tuple also a quintuple, a 6-tuple also a sextuple. The series is continued analogously by Latin multiplication number words. The 0-tuple is called empty tuple and is ()notated by


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