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The axiom of extensionality, [1] [2] also called the axiom of extent, [3] [4] is an axiom used in many forms of axiomatic set theory, such as Zermelo–Fraenkel set theory. [5] [6] The axiom defines what a set is. [1] Informally, the axiom means that the two sets A and B are equal if and only if A and B have the same members.
Thus, the axiom of the empty set is implied by the nine axioms presented here. The axiom of extensionality implies the empty set is unique (does not depend on ). It is common to make a definitional extension that adds the symbol "" to the language of ZFC.
In set theory, the axiom of extensionality states that two sets are equal if and only if they contain the same elements. In mathematics formalized in set theory, it is common to identify relations—and, most importantly, functions —with their extension as stated above, so that it is impossible for two relations or functions with the same ...
The axioms of Zermelo set theory are stated for objects, some of which (but not necessarily all) are sets, and the remaining objects are urelements and not sets. Zermelo's language implicitly includes a membership relation ∈, an equality relation = (if it is not included in the underlying logic), and a unary predicate saying whether an object is a set.
General set theory (GST) is George Boolos's (1998) ... Axiom of Extensionality: The sets x and y are the same set if they have the same members.
To understand this axiom schema, note that the set B must be a subset of A. Thus, what the axiom schema is really saying is that, given a set A and a predicate , we can find a subset B of A whose members are precisely the members of A that satisfy . By the axiom of extensionality this
Together with the axiom of choice (see below), these are the de facto standard axioms for contemporary mathematics or set theory. They can be easily adapted to analogous theories, such as mereology. Axiom of extensionality; Axiom of empty set; Axiom of pairing; Axiom of union; Axiom of infinity; Axiom schema of replacement; Axiom of power set ...
If any set is postulated to exist, such as in the axiom of infinity, then the axiom of empty set is redundant because it is equal to the subset {}.Furthermore, the existence of a member in the universe of discourse, i.e., ∃x(x=x), is implied in certain formulations [1] of first-order logic, in which case the axiom of empty set follows from the axiom of Δ 0-separation, and is thus redundant.