TITLE: Difference between equality and isomorphism
QUESTION [14 upvotes]: I don't really understand the difference between sets being equal and isomorphic.  I know that two sets $A$ and $B$ are equal of $A \subseteq B$ and $B \subseteq A$.  I know two sets are isomorphic is there is a homomorphic bijection between the two sets, i.e., they have the same cardinality and the elements of each can be identified in such a way that they behave in exactly the same way.  (In other words, the two sets have the same structure.)
Is equality stronger?  I guess my main question is:
If two sets $A$ and $B$ are equal, does this imply they are isomorphic?  How do $A \subseteq B$ and $B \subseteq A$, which are purely set theoretic statements, imply that the structure of the two sets is the same, i.e., their elements can be identified in such a way that they act the same way?

REPLY [8 votes]: Isomorphism means "similar structure", so you have to be more accurate and say what type of structure. If you want to talk about every possible structure, then only way two sets are "always isomorphic" is if they are both the empty set (which carries no structure other than empty sets).
Equality as sets, as Henning writes in his excellent answer, means that the sets are in fact equal. But note that when we say that the group $G$ is equal to the group $H$ we mean not only that the sets are equal, but also that the operation is equal (as a set of ordered triplets, or so). We can easily define two non-isomorphic group structures on an infinite set.
If you only consider them as unstructured sets, then isomorphism just says that there is a bijection between the two sets; but then you can't say that the elements "act the same way" since there are no actions. Equality means that the two sets are the same set. Note that $\{\varnothing\}$ and $\{\{\varnothing\}\}$ are both singletons, and are therefore isomorphic as sets (i.e. they have the same cardinality), but they are certainly not equal.