1. equivalence classes

Hi i cans show that the below problem is an equivalence class no problem, but i am finding it difficult to describe its equivalence classes.

2. The post says "whenever $|A|-|B|$." What does that mean?

3. Originally Posted by Plato
The post says "whenever $|A|-|B|$." What does that mean?
It says $A \sim B$ whenever $|A| = |B|$

4. Originally Posted by Jhevon
It says $A \sim B$ whenever $|A| = |B|$
Not in the image that I see.

5. Originally Posted by Plato
Not in the image that I see.
Well, I don't know what's going on. That's what I see.

6. Originally Posted by Jhevon
Well, I don't know what's going on. That's what I see.
I think that is why we ought to insist on the use of LaTeX.

7. Originally Posted by Plato
I think that is why we ought to insist on the use of LaTeX.
Maybe. Or at least insist that questions are not posted in image files, unless there are accompanying diagrams or something.

Originally Posted by 1234567
Hi i cans show that the below problem is an equivalence class no problem, but i am finding it difficult to describe its equivalence classes.

I don't really see a better way to describe the classes other than to reuse the language of the problem. Something like,

For $A \in \mathcal P (\mathbb N),~ [A] = \{ B \in \mathcal P (\mathbb N) ~:~ |A| = |B| \}$

8. Originally Posted by Jhevon
Maybe. Or at least insist that questions are not posted in image files, unless there are accompanying diagrams or something.

I don't really see a better way to describe the classes other than to reuse the language of the problem. Something like,

For $A \in \mathcal P (\mathbb N),~ [A] = \{ B \in \mathcal P (\mathbb N) ~:~ |A| = |B| \}$
What about saying it a little better. The equivalence class of a subset of the naturals under this relation is the class of all sets such that there exists a bijection between that set and the class representative.

9. Originally Posted by Drexel28
What about saying it a little better. The equivalence class of a subset of the naturals under this relation is the class of all sets such that there exists a bijection between that set and the class representative.
that is fine i suppose. i wanted to emphasize that we are dealing with sets here. and what you described is what |A| = |B| means by definition. so it's a matter of taste, i think... which i think is also what you're saying.

10. Originally Posted by Jhevon
that is fine i suppose. i wanted to emphasize that we are dealing with sets here. and what you described is what |A| = |B| means by definition. so it's a matter of taste, i think... which i think is also what you're saying.
It is just a matter of taste haha.

11. A~B iff A and B have the same cardinal.
therefore, there are countable many equivalent classes:
1-class: the collection of all sets that contains only one element.
2-class: the collection of all sets that contains two elements.
...
n-class: the collection of all sets that contains n elements.
...
infinite-class:the collection of all sets that contains countable infinitely many elements.

A further question: What is P(N)/~? I leave it for you to solve it. Solve it, and you will understand the cardinality better.