In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of pairing is one of the axioms of Zermelo-Fraenkel set theory. Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. ... Logic (from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of arguments, although the exact definition of logic is a matter of controversy amongst philosophers (see below). ... Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Wikibooks Wikiversity has more about this subject: School of Computer Science Open Directory Project: Computer Science Downloadable Science and Computer Science books Collection of Computer Science Bibliographies Belief that title science in computer science is inappropriate Category: Computer science ... In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. ... The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ...

In the formal language of the Zermelo-Frankel axioms, the axiom reads: In mathematics, logic and computer science, a formal language is a set of finite-length words (i. ...

or in words:

Given any set A and any set B, there is a set C such that, given any set D, D is a member of C if and only if D is equal to A or D is equal to B.

What the axiom is really saying is that, given two sets A and B, we can find a set C whose members are precisely A and B. We can use the axiom of extensionality to show that this set C is unique. We call the set C the pair of A and B, and denote it {A,B}. Thus the essence of the axiom is: In predicate logic, universal quantification is an attempt to formalise the notion that something (a logical predicate) is true for everything, or every relevant thing. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... there is is a song by boxcar racer ... ↔ ⇔ ≡ logical symbols representing iff. ... See also the disambiguation page title equality. ... OR logic gate In mathematics, logical disjunction (usual symbol or) is a logical operator that results in true if either of the operands is true. ... In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo-Fraenkel set theory. ...

Any two sets have a pair.

{A,A} is abbreviated {A}, called the singleton containing A. Note that a singleton is a special case of a pair. In mathematics, a singleton is a set with exactly one element. ...

The axiom of pairing also allows for the definition of ordered pairs. For any sets a and b, the ordered pair is defined by the following: An ordered pair is a collection of two objects such that one can be distinguished as the first element and the other as the second element. ... An ordered pair is a collection of two objects such that one can be distinguished as the first element and the other as the second element. ...

Note that this definition satisfies the condition

Ordered n-tuples can be defined recursively as follows: In mathematics, a tuple is a finite sequence of objects, that is, a list of a limited number of objects (an infinite sequence is a family). ...

The axiom of pairing is generally considered uncontroversial, and it or an equivalent appears in just about any alternative axiomatization of set theory. Nevertheless, in the standard formulation of the Zermelo-Fraenkel set theory, the axiom of pairing follows from the axiom of power set and the schema of replacement, thus it is sometimes omitted. In mathematics, axiomatization is the process of defining the basic axiomatic systems from which mathematical theories can be derived. ... The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ... In mathematics, the axiom of power set is one of the Zermelo-Fraenkel axioms of axiomatic set theory. ... In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom schema of replacement is a schema of axioms in Zermelo-Fraenkel set theory. ...

Generalisation

Together with the axiom of empty set, the axiom of pairing can be generalised to the following statement: In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of empty set is one of the axioms of Zermelo_Fraenkel set theory. ...

that is:

Given any finite number of sets A₁ through A_n, there is a set C whose members are precisely A₁ through A_n.

This set C is again unique by the axiom of extension, and is denoted {A₁,...,A_n}. In mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ...

Of course, we can't refer to a finite number of sets rigorously without already having in our hands a (finite) set to which the sets in question belong. Thus, this is not a single statement but instead a schema, with a separate statement for each natural number n. The word schema comes from the Greek word σχήμα (skhēma) that means shape or more generally plan. ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), and they can be used for ordering (this is...

• The case n = 1 is the axiom of pairing with A = A₁ and B = A₁.
• The case n = 2 is the axiom of pairing with A = A₁ and B = A₂.
• The cases n > 2 can be proved using the axiom of pairing and the axiom of union multiple times.

For example, to prove the case n = 3, use the axiom of pairing three times, to produce the pair {A₁,A₂}, the singleton {A₃}, and then the pair {{A₁,A₂},{A₃}}. The axiom of union then produces the desired result, {A₁,A₂,A₃}. We can extend this schema to include n=0 if we interpret that case as the axiom of empty set. In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of union is one of the axioms of Zermelo_Fraenkel set theory, stating that, for any two sets, there is a set that contains exactly the elements of both. ... In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of empty set is one of the axioms of Zermelo_Fraenkel set theory. ...

Thus, one may use this as an axiom schema in the place of the axioms of empty set and pairing. Normally, however, one uses the axioms of empty set and pairing separately, and then proves this as a theorem schema. Note that adopting this as an axiom schema will not replace the axiom of union, which is still needed for other situations. In symbolic logic, it is sometimes inconvenient or impossible to express an axiomatic system in a finite number of axioms. ... A theorem is a proposition that has been or is to be proved on the basis of explicit assumptions. ... In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of union is one of the axioms of Zermelo_Fraenkel set theory, stating that, for any two sets, there is a set that contains exactly the elements of both. ...