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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. It was introduced by Zermelo (1908) as a special case of his axiom of elementary sets.
theory and the branches of logic, mathematics, and computer science that use it, the axiomofpairing is one of the axiomsof Zermelo–Fraenkel set theory...
Axiomof extensionality Axiomof empty set AxiomofpairingAxiomof union Axiomof infinity Axiom schema of replacement Axiomof power set Axiomof regularity...
of mathematics and philosophy that use it, the axiomof infinity is one of the axiomsof Zermelo–Fraenkel set theory. It guarantees the existence of at...
the axiomof choice, abbreviated AC or AoC, is an axiomof set theory equivalent to the statement that a Cartesian product of a collection of non-empty...
comprehension, or the axiomof regularity and axiomofpairing. In Zermelo–Fraenkel set theory, the axiomof regularity and axiomofpairing prevent any set...
theory, the existence of unordered pairs is required by an axiom, the axiomofpairing. More generally, an unordered n-tuple is a set of the form {a1, a2,...
Together with the axiomofpairing, this implies that for any two sets, there is a set (called their union) that contains exactly the elements of the two sets...
an axiom is a premise or starting point for reasoning. In mathematics, an axiom may be a "logical axiom" or a "non-logical axiom". Logical axioms are...
See Axiomofpairs. AXIOM III. Axiomof separation (Axiom der Aussonderung) "Whenever the propositional function –(x) is defined for all elements of a set...
(See axiomofpairing.) Note the following points: The order of elements is immaterial; for example, {1, 2} = {2, 1}. Repetition (multiplicity) of elements...
set theory, the axiom schema of replacement is a schema ofaxioms in Zermelo–Fraenkel set theory (ZF) that asserts that the image of any set under any...
of axiomatic set theory, the axiom schema of specification, also known as the axiom schema of separation (Aussonderung Axiom), subset axiom or axiom schema...
Axiom Space, Inc., also known as Axiom Space, is an American privately funded space infrastructure developer headquartered in Houston, Texas. Founded in...
and no new elements were added, this is the empty set of L {\displaystyle L} . Axiomofpairing: If x {\displaystyle x} , y {\displaystyle y} are sets...
In mathematics, the axiomof determinacy (abbreviated as AD) is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962...
than the class of all sets Axiomofpairing Unordered pairsof sets are sets Axiomof power set The powerset of any set is a set Axiomof projective determinacy...
understand the role of ordered pairs in mathematics. Hence the ordered pair can be taken as a primitive notion, whose associated axiom is the characteristic...
foundational system for the whole of mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiomof choice. Besides its foundational...
an axiom schema (plural: axiom schemata or axiom schemas) generalizes the notion ofaxiom. An axiom schema is a formula in the metalanguage of an axiomatic...
theories, the axiomof global choice is a stronger variant of the axiomof choice that applies to proper classes of sets as well as sets of sets. Informally...
definition of "number" uses an axiomof that system – the axiomofpairing – that leads to the definition of "ordered pair" – no overt number axiom exists...
existence of the unordered pair is given by the AxiomofPairing, the existence of the empty set follows by Separation from the existence of any set, and...
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