How does the axiom of extensionality determine when two sets are equal?

The axiom of extensionality establishes that a set is determined solely by its members. Under this principle, two sets are considered equal if and only if they contain precisely the same elements.123

This means that identity between sets is defined by identity of membership rather than by how the sets are described or specified. Even if two sets are defined by different conditions—for example, two different formulas generating the same collection of objects—they are regarded as the same set if their members are identical. Consequently, if every member of set $A$ is a member of set $B$, and every member of set $B$ is a member of set $A$, then $A=B$.32