Pair set

As a pair of quantity , two quantity or pair is referred to in the set theory by symbolized amount exactly the objects and as elements contains. The following applies: ${\displaystyle \{a,b\}}$ ${\displaystyle a}$ ${\displaystyle b}$

${\displaystyle z\in \{a,b\}\;\iff \;z=a\lor z=b}$.

In the older, naive set theory , which was not yet axiomatized , the existence of a set described by extensional enumeration was intuitively justified. In axiomatic set theory, however, since the Zermelo set theory of 1907, the existence of pair sets has been required by a pair set axiom. This axiom was adopted in all important set theories, for example in the Zermelo-Fraenkel set theory ZF or the Neumann-Bernays-Gödel set theory NBG. This pair set axiom reads in verbal form: For all A and B there is a set C which has exactly A and B as elements. In predicate logic it reads:

${\displaystyle \forall A,B\colon \exists C\colon \forall D\colon (D\in C\iff (D=A\lor D=B))}$

The couple amount axiom in ZF and NBG but a redundant axiom, because there may be from the other axioms follows are derived: Take the empty set by empty set axiom , twice is the power set by power set axiom and thus receives the special couple crowd whose elements by replacing axiom by any other elements can be replaced. In the older Zermelo set theory without Fraenkel's substitution axiom from 1921, this derivation was still impossible. ${\displaystyle \{\emptyset ,\{\emptyset \}\}}$

The set required in the pair set axiom is unique due to the axiom of extensionality and is noted in the form given above. The pair set axiom says nothing about the kind of elements. The objects can vary, depending on the selected set theory. In the context of ZF and NBG, which both represent a pure set theory, there are only sets, in a set theory with primordial elements it can also be such, for example in ZFU .

An additional axiom for the singular set or singular set is not required. Because the set does not necessarily have to contain two different elements. In this case, there is only a single element set, since elements in sets are not counted twice. Likewise, no axiom is necessary for larger sets obtained by enumeration, because larger finite sets are obtained successively via the union axiom . All these sets with an extensional enumeration of the elements are thus defined: ${\displaystyle \,\{a,b\}}$ ${\displaystyle \,a=b}$

${\displaystyle \,\{a\}:=\{a,a\}}$,

${\displaystyle \{a,b,c\}:=\{a,b\}\cup \{c\}}$

${\displaystyle \{a,b,c,d\}:=\{a,b,c\}\cup \{d\}}$

and so on.

Different meaning

Sometimes the term "pair set" is also used in the sense of a set of pairs for the Cartesian product of two sets.

literature

• Oliver Deiser: Introduction to set theory. Georg Cantor's set theory and its axiomatization by Ernst Zermelo. 2nd, improved and enlarged edition. Springer, Berlin et al. 2004, ISBN 3-540-20401-6 .