Thread: Equivalence classes

\Rightarrow "Let be an equivalence relation on . For any either



or

"

My professor gave the following proof to the preceding lemma in his notes, but he starts it by saying "It is enough to assume" then footnoting this and writing "no it isn't". So how do I need to add to this to make it a complete proof?

It is enough to assume (footnoted here)

and deduce that


First, fix For any ,

means
(similarly)
(symmetry)
Therefore (transitivity), i.e.,


That is, for every

In other words,



In particular,
and also

Therefore

So, if choose some

Then and , so

Originally Posted by Conn

\Rightarrow "Let be an equivalence relation on . For any either



or

"

My professor gave the following proof to the preceding lemma in his notes, but he starts it by saying "It is enough to assume" then footnoting this and writing "no it isn't". So how do I need to add to this to make it a complete proof?

It is enough to assume (footnoted here)

and deduce that


First, fix For any ,

means
(similarly)
(symmetry)
Therefore (transitivity), i.e.,


That is, for every

In other words,



In particular,
and also

Therefore

So, if choose some

Then and , so

It is complete. What you want to show is that given two equivalence classes they're equal or disjoint? You've showed that their not disjoint implies their equal and so either they are equal, or they're not equal in which case they must be disjoint (otherwise they'd be equal!).