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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.
GST features a single primitive ontological notion, that of set, and a single ontological assumption, namely that all individuals in the universe of discourse (hence all mathematical objects) are sets. There is a single primitive binary relation, set membership; that set a is a member of set b is written a ∈ b (usually read "a is an element ...
Adopting the standard definition of set equality via extensionality, the full Axiom of Choice is such a non-constructive principle that implies for the formulas permitted in one's adopted Separation schema, by Diaconescu's theorem. Similar results hold for the Axiom of Regularity existence claim, as shown below.
By the axiom of extensionality, class in the intersection axiom and class in the complement and domain axioms are unique. They will be denoted by: A ∩ B , {\displaystyle A\cap B,} ∁ A , {\displaystyle \complement A,} and D o m ( A ) , {\displaystyle Dom(A),} respectively.
The proof below is therefore given using the means of a constructive set theory. It is evident from the proof how the theorem relies on the axiom of pairing as well as an axiom of separation, of which there are notable variations. A crucial role in the set theoretic proof is also played by the axiom of extensionality. The subtleties the latter ...