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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.

  1. What does "well-defined as a set" mean? Does it mean $A\cup B$ need not always have the very same elements?

  2. 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?

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What textbook are you using? Can you give some context? – Keshav Srinivasan Mar 15 '14 at 15:05
@KeshavSrinivasan- I'm using "Chapter 0" by Aluffi. – algebraically_speaking Mar 18 '14 at 6:36

I'm not quite sure about the context in which this is said, but I'd say that "$A\cup B$ is not well-defined as a set but it is well defined up to isomorphism" means that if you replace $A$ and $B$ with isomorphic objects $A^\prime$ and $B^\prime$ then $A^\prime\cup B^\prime$ doesn't contain exactly the same elements of $A\cup B$, but it is nonetheless isomorphic to it.

A bit more of details about the occurence of this, might greatly help clarifying it.

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