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Math Help - smaller set: S/~ ?

  1. #1
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    Question smaller set: S/~ ?

    On p. 17 of "Applied Differential Geometry" by William Burke, he uses the idea of a "smaller set." (AxA)/~. He initially mentioned this idea, only briefly, on p. 8:

    "The smaller set formed from the set S by using the equivalence relation ~ is written S/~."

    So what is the smaller set? (google and wikipedia have no info)

    Thanks in advance.
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  2. #2
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    S/~ has the members of equivalent classes using ~ as the equivalence relation. Typically called quotient space.
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  3. #3
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    Quote Originally Posted by xxp9 View Post
    S/~ has the members of equivalent classes using ~ as the equivalence relation. Typically called quotient space.
    Sorry for the late reply. Abstract math was never my forte' (as you can see by my mundane example below) and so there was much for me to absorb. But I think I understand it now.

    If I understand correctly:

    Let X := {a, A, \alpha, b, B, \beta, g, G, \gamma}

    Then:
    [a] is the equivalence class: {a, A, \alpha},
    [b] is the equivalence class: {b, B, \beta}, and
    [g] is the equivalence class: {g, G, \gamma}.

    Thus the quotient set of X using the equivalence relation '~' (which reads: "all letters similar to the first") is then:

    X/~ = {[a], [b], [g]}

    And so this is what Burke refers to as "the smaller set."
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  4. #4
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    even I don't what the author meant but your understanding is correct for quotient space.
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  5. #5
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    Smile

    Quote Originally Posted by xxp9 View Post
    even I don't what the author meant but your understanding is correct for quotient space.
    Now that you mention it I suppose I have to impose my own understanding of what he meant by "smaller set."

    From what I have read so far, he had a powerful intuitive understanding of the math involved in physics - as well as the physics. He also points out the failings of physicists' lack of understanding of the math (in a constructive way).

    E.g. he said that much of physics uses affine space. The equivalence of vectors is translated as vectors are allowed to be transposed all over the space. When he said that I finally understood free vectors, and affine space not having an origin.

    So I am sure his meaning is not a simple renaming of a quotient space - I'll just have to think about it some more.

    Thanks for the reply and letting me know I finally understand something in abstract (aka modern) math!
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