How did the axiom of extensionality determine when two sets are the same?

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logic: universal affirmative — Modern logic Representations of the universal affirmative, “All A's are B's” in modern logic.

In the context of modern logic and set theory, the axiom of extensionality establishes that a set is determined solely by its members.1

Under this principle, two sets are considered identical if and only if they contain precisely the same members. This means that the identity of a set is defined by its membership rather than by the specific conditions or descriptions used to specify it. For example, even if two sets are defined by different properties, they are regarded as the same set if every element belonging to the first also belongs to the second, and vice versa.2

This principle is fundamental to the axiomatic development of set theory, as it provides a clear criterion for equality that distinguishes the identity of classes from the methods used to define them.12