My Algebra textbook says the following:
$A\cup B$ is defined as the union of $A'$ and $B'$, where $A'$ and $B'$ are isomorphic to $A$ and $B$ respectively. Hence, $A\cup B$ is not well-defined as a set, but it is well defined up to isomorphism.
What does "well-defined as a set" mean? Does it mean $A\cup B$ need not always have the very same elements?
What does "well defined up to isomorphism" mean? Does it mean that the set contains the very same elements if we consider isomorphic elements to be the same?