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 $A/\sim$, a set A with an equivalence relation $\sim$ such that the set of equivalence classes is uncountable and the equivalence classes contain an uncountable amount of elements.
I already tried with $A=\mathbb{R}$ and $x\sim y:=x-y\in \mathbb{Z}$. The equivalence classes are the sets of reals with the same fractional part. You can show that $A/\sim =\left[0,1\right)$. Which is easy to show is uncountable with an injection $f:\left\{0,1{\right\}}^{\mathrm{\infty }}\to \left[0,1\right)$ (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 ${\mathbb{R}}^{2}$, define $\left(x,y\right)\sim \left(a,b\right)$ iff $x=a$. Equivalence classes are copies of $\mathbb{R}$, and there are $\mathbb{R}$ of them (one per $x\in \mathbb{R}$).
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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 $\left\{{A}_{i}{\right\}}_{i\in I}$ be a collection of uncountable sets. Define the $A:=\left\{\left(a,i\right):i\in I,a\in {A}_{i}\right\}.$.
Step 2
In other words, the above is the disjoint union ${\bigsqcup }_{i\in I}{A}_{i}$. Define the relation $\sim$ on A by $\left(a,i\right)\sim \left(b,j\right)⇔i=j.$.
Then, the equivalence classes are precisely the copies of ${A}_{i}$, i.e., $A/\sim =\left\{{A}_{i}×\left\{i\right\}:i\in I\right\}$.
In some sense, this is the most general way of getting an uncountable set of uncountable equivalence classes.
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