Uncountable set of uncountable equivalence classes I am trying to find A <mrow class="MJX-Te

ziphumulegn 2022-07-15 Answered
Uncountable set of uncountable equivalence classes
I am trying to find A / , a set A with an equivalence relation such that the set of equivalence classes is uncountable and the equivalence classes contain an uncountable amount of elements.
I already tried with A = R and x y := x y Z . The equivalence classes are the sets of reals with the same fractional part. You can show that A / ∼= [ 0 , 1 ). Which is easy to show is uncountable with an injection f : { 0 , 1 } [ 0 , 1 ) (this is the standard method I use to show a set is uncountable).
However, I think all elements in my equivalence classes are countable, as they are all natural numbers plus a specific fractional part. Am I right to think like this, does it make my example false ? In case it is false, is there a way to "fix" it?
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Answers (2)

Zackery Harvey
Answered 2022-07-16 Author has 21 answers
Explanation:
In R 2 , define ( x , y ) ( a , b ) iff x = a. Equivalence classes are copies of R , and there are R of them (one per x R ).
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Sylvia Byrd
Answered 2022-07-17 Author has 6 answers
Step 1
The other answers have already given concrete examples. Here's something more abstract that you can do. Let I be any uncountable set and let { A i } i I be a collection of uncountable sets. Define the A := { ( a , i ) : i I , a A i } ..
Step 2
In other words, the above is the disjoint union i I A i . Define the relation on A by ( a , i ) ( b , j ) i = j ..
Then, the equivalence classes are precisely the copies of A i , i.e., A / = { A i × { i } : i I }.
In some sense, this is the most general way of getting an uncountable set of uncountable equivalence classes.
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