Axiom of extensionality information

of logic, mathematics, and computer science that use it, the axiom of extensionality, axiom of extension, or axiom of extent, is one of the axioms of...

, the power set of x {\displaystyle x} , consisting precisely of the subsets of x {\displaystyle x} . By the axiom of extensionality, the set P ( x )...

Axiom of extensionality Axiom of empty set Axiom of pairing Axiom of union Axiom of infinity Axiom schema of replacement Axiom of power set Axiom of regularity...

B)\to A=B.} This axiom is identical to the axiom of extensionality found in many other set theories, including ZF. Any element or a subset of a set is a set...

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 logic, extensionality, or extensional equality, refers to principles that judge objects to be equal if they have the same external properties. It stands...

Zermelo–Fraenkel axioms (but not the axiom of extensionality, the axiom of regularity, or the axiom of choice) then became necessary to make up for some of what was...

assume any axioms except the axiom of extensionality and the axiom of induction—a natural number is either zero or a successor and each of its elements...

the axiom of choice, abbreviated AC or AoC, is an axiom of set theory equivalent to the statement that a Cartesian product of a collection of non-empty...

interpreted in a weak set theory whose axioms are extensionality, the existence of the empty set, and the axiom of adjunction (Tarski 1953, p.34). In fact...

In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every non-empty...

predicate. AXIOM I. Axiom of extensionality (Axiom der Bestimmtheit) "If every element of a set M is also an element of N and vice versa ... then M ≡...

words: There is a set such that no element is a member of it. We can use the axiom of extensionality to show that there is only one empty set. Since it is...

Look up extension, extend, or extended in Wiktionary, the free dictionary. Extension, extend or extended may refer to: Axiom of extensionality Extensible...

existence of the empty set is assured by the axiom of empty set, and its uniqueness follows from the axiom of extensionality. However, the axiom of empty...

form. In particular the equivalence holds in the presence of the axioms of extensionality, pairing, union and powerset. ∀ A ( [ ∀ x ∃ ! y ϕ ( x , y ...

that is, if every element of A is an element of B and every element of B is an element of A. (See axiom of extensionality.) Thus a set is completely...

the axiom of extensionality must be formulated to apply only to objects that are not urelements. This situation is analogous to the treatments of theories...

an axiom schema (plural: axiom schemata or axiom schemas) generalizes the notion of axiom. An axiom schema is a formula in the metalanguage of an axiomatic...

are using the same element relation and no new sets were added. Axiom of extensionality: Two sets are the same if they have the same elements. If x {\displaystyle...

theories, the axiom of global choice is a stronger variant of the axiom of choice that applies to proper classes of sets as well as sets of sets. Informally...

The axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set is constructible. The axiom is usually written...