Tuple

Tuple stand next to quantities ordered pairs and functions in the center of the mathematical concept of the world, they are in many disciplines to the building blocks of their structures. A tuple is a finite list, are given in the one behind the other mathematical objects that are not necessary to be different from each other, the order is important. For example, the lists and two consisting of zeros and ones of the tuple Tuple Length 5 of the length are called tuples and in particular 2 -tuple, 3 -tuple, 4 -tuple, 5-tuple, ... also ordered in pairs, triples, quadruple, quintuple, and so on (hence the name " tuple ").

Definitions and notations

The standing in the first place a non-empty tuple object is called its first component, analogue one speaks of his second, third, etc. component and refers to these means of projection operators. As you can tuple explicitly write down show examples:

Examples in horizontal notation

• If = 0

In the last example, the components of the 3- tuple without brackets and without delimiters are listed.

Examples in vertical notation

Tuples, the tuples are component ( Matrix notation )

Equality of tuples

Two tuples are valid if and only considered equal if they have the same length and their corresponding components are equal. Formal ( the operator denotes the length of a tuple ):

View as quantity

As any mathematical object, including tuple can be represented as volumes. Here are two examples:

• This representation is a generalization of a representation of ordered pairs as a set.

• ( ordered pairs of square brackets). With this representation, tuples are defined as finite sequences.

Tuples as nested ordered pairs

In addition to the very descriptive definition of tuple as a list of mathematical objects can be found in the literature often yet another, in which no empty tuple (0 -tuple ) is defined, any mathematical object is a 1- tuple and each ordered pair with an n- tuple as a linker moiety (n 1) - tuple is:

These tuples have the characteristic that each n - tuple ( n> 1) and an (n - 1) - tuples, so that this does not define the length of a tuple, and therefore can not also from the i-th component of a tuple speak. One speaks here rather of the "i- th component of the aspect that an n-tuple is " and recorded it with the projection operator.

Example: For being.

The same applies to the demand for equality of tuples: Considering the fact that two tuples are equal, they are exactly then considered equal if their corresponding components are equal. Formal:

This Tupelbegriff also allows proper classes as components, as ordered pairs are defined for real classes.

Use

Tuples are used in mathematics, for example, as coordinates of points or vectors in - dimensional spaces and in the computer science as data fields and structures. Also, rows and columns as well as the matrices tuple ( a tuple is a matrix whose components are tuples. )