Ordered pair

An ordered pair, also called 2- tuple is in mathematics is an important way to combine two not necessarily different from each other mathematical objects. In this case one of the two objects is excellent and is called while the other object is called right, second or rear component of the couple left, first or front component of the pair. Listed is an ordered pair by its components, separated by a comma, after another writes, the left first, and the whole thing including in a suitable pair of parentheses, usually the round. Ordered pairs are central to the mathematical concept of the world and are the basis of many building blocks of more complex mathematical objects.

Equality of ordered pairs

The term of the ordered pair is characterized by Peano axiom pair:

With the two projection operators to give the left respectively right component of an ordered pair as and. This allows the couple formalize axiom:

Representation of ordered pairs as sets

In the literature, there are, among others for the ordered pair following representations as sets:

  • For tuple - term generalizable representation
  • , According to Norbert Wiener ( 1914)
  • , Mainstream account to Kazimierz Kuratowski ( 1921). A variant is the definition
  • So -called short presentation
  • , With each other and are different objects, both also different from and, after Felix Hausdorff (1914 )
  • , According to Jürgen Schmidt (1966 ) with reference to Quine; may also be true classes here.

Using ordered pairs

Ordered pairs are the basic building blocks of many mathematical structures. For example,

  • Defined in set theory, Cartesian products, relations, functions, sequences as sets of ordered pairs,
  • In analysis complex numbers as ordered pairs of real numbers as components, real numbers as sets (equivalence classes) of infinite sequences ( Cauchy sequences of rational numbers ), rational numbers as equivalence classes of ordered pairs whose components are integers, integers as equivalence classes of ordered pairs, the components are natural numbers, defined
  • The algebraic structures, for example, groups, rings, fields defined essentially in algebra as functions ( binary logic ).