axiom of extensionality word definition

Question

I was reading about the axiom of extensionality and in words it reads "If A and B are sets such that for every element x, x is a member of A if and only if x is a member of B, then A is equal to B" am i right in saying that this is not actually saying that A and B have precisely the same members just that A is a subset of B? Thanks

Answer 1

Note that it is "if and only if". That means: if $x\in A$ then $x\in B$ and also if $x\in B$ then $x\in A$. So $A$ must be a subset of $B$ and also $B$ must be a subset of $A$. The words "if and only if" always mean there are two directions in the statement.