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Math Help - equivalence classes

  1. #1
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    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.

    equivalence classes-untitled.jpg
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    The post says "whenever |A|-|B|." What does that mean?
    That is not a relation. Please reply with a complete question.
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Plato View Post
    The post says "whenever |A|-|B|." What does that mean?
    That is not a relation. Please reply with a complete question.
    It says A \sim B whenever |A| = |B|
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    Quote Originally Posted by Jhevon View Post
    It says A \sim B whenever |A| = |B|
    Not in the image that I see.
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Plato View Post
    Not in the image that I see.
    Well, I don't know what's going on. That's what I see.
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    Quote Originally Posted by Jhevon View Post
    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.
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Plato View Post
    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.

    Quote Originally Posted by 1234567 View Post
    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.

    Click image for larger version. 

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    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| \}
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Jhevon View Post
    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.
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Drexel28 View Post
    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.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Jhevon View Post
    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.
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  11. #11
    Senior Member Shanks's Avatar
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    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.
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