is the equipotent relation an equivalence relation?

Question

I know that the relation satisfy for $A,B,C$ sets, the following conditions. $A$~$A$, if $A$~$B$ then $B$~$A$ and, if $A$~$B$ and $B$~$C$ then $A$~$C$. but not necesary exist a set U such that $A,B,C \subset U$.

Can someone explains me?

Answer 1

The hard facts first:

The equipotency relation satisfies all the properties we require of equivalence relations, except that it actually doesn't exist as a set of pairs.

The exception doesn't really have anything to do with the concept of "equivalence relation" specifically. It's probably fair to say that if you recognize it as a "relation" at all, then you have no reason not to call it an "equivalence relation" too.

Whether you should care about not being a set depends on what your goal is.

The relations we meet in ordinary relations can pretty universally be represented as sets, so there the question of whether it's a set never really arises and we can go directly to checking the usual conditions. Only within set theory itself do we routinely encounter relations that are so large that it cannot be sets.*

In plenty of contexts this is not a problem, and it is much more useful to think of the relation that it satisfies the three usual conditions and therefore we can use the same intuition about it as we have already developed about equivalence relations in, say, abstract algebra. For the purpose of guiding your intuition, equipotency should without doubt be considered an equivalence relation.

But it should be kept in mind that the formal consequences of being an equivalence relation are not guaranteed here. For example, the equivalence classes are not sets.

*Actually not quite true. There are similar problems about things such as the "isomorphic to" relation between groups or other algebraic structures, which also is an equivalence relation but for not being a set.

Answer 2

Well, I think yes there does exist a set that contains $A, B, C$ and the set is $A\cup B \cup C$, and this is true for any arbitrary collection of sets.

Answer 3

It seems your problem is more philosophical than mathematical

• If you want to consider equivalence relations in a given model for set theory (ZFC is the standard one) then it doesn't make sense to define a relation between arbitrary sets;
• If you want to work with something that "look and feel like sets" but want to be able to talk about relations with sets, you should use "Classes" instead of "Sets". See for example vNGB. In this case we can talk about the class of all sets, which is not itself a set.
• If you want to avoid any of this and just work on a purely formal level, you can think you are doing First order calculus on the usual Formal language of mathematics, so you define a new binary symbol and prove that it satisfies the axioms for an equivalence relation at a formal (symbolic) level.

The usual approach is... don't think too much about it. Of course you'd want to define a relation between sets - equipotency is really natural - so the first approach works only if you restrict yourself to sets which are all contained in a big "universe" set $U$. But in this case you can't compare your big set $U$ with any of the sets if contains. If you want to do that, you need to take the power set $P(U)$ of $U$, but you have the same problem again: *can't compare $P(U)$, in principle, with the other sets...

In general, you keep the second approach in mind (think of a "collection of all sets", whathever that is), but prove it formally, with the third approach, as you did.