Axiom of extensionality

In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo–Fraenkel set theory.

In the formal language of the Zermelo–Fraenkel axioms, the axiom reads:

or in words:

The converse, ${\displaystyle \forall A\,\forall B\,(A=B\Rightarrow \forall X\,(X\in A\iff X\in B))}$, of this axiom follows from the substitution property of equality.

To understand this axiom, note that the clause in parentheses in the symbolic statement above simply states that A and B have precisely the same members. Thus, what the axiom is really saying is that two sets are equal if and only if they have precisely the same members. The essence of this is:

The axiom of extensionality can be used with any statement of the form ${\displaystyle \exists A\,\forall X\,(X\in A\iff P(X)\,)}$, where P is any unary predicate that does not mention A, to define a unique set ${\displaystyle A}$ whose members are precisely the sets satisfying the predicate ${\displaystyle P}$. We can then introduce a new symbol for ${\displaystyle A}$; it's in this way that definitions in ordinary mathematics ultimately work when their statements are reduced to purely set-theoretic terms.

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