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Math Help - function and relation

  1. #1
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    function and relation

    The examples of equivalence classes in my book are mostly related to integers. I would like to know this:

    Can a continuous function have equivalence classes?
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  2. #2
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    What conditions need to be satisfied for a continuous function to be:

    1) A well-defined function
    2) An equivalence relation?

    Answer those and you will get your result; note that you're not even considering the continuity.
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  3. #3
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    Do you mind showing a simple example for someone like me?
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  4. #4
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    Sure:

    Any function is a relation. Specifically, a function f:A \to B is a relation f \subseteq A \times B, such that for any a \in A, there is exactly one b \in B such that (a,b) \in f.

    Now, for f to be an equivalence relation, its domain has to be its range - f: A \to A.

    Specifically, it also has to be reflexive. That is, for each a \in A, \ (a,a) \in f.

    What does that tell you about f?
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  5. #5
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    Very interesting. It looks like an identity function, but what does the equivalence classes look like since the points are so dense?
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  6. #6
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    Oh, yeah, you already said about the continuity being irrelevant. Thank you for your valuable time.
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  7. #7
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    Well, what I meant was that you can get the result without assuming continuity, but the function you get (identity function) is always continuous. The equivalence class of x \in \mathbb{R} is [x] = \{ y \in \mathbb{R} : (x,y) \in f \}  = \{ y \in \mathbb{R} : f(x)=y \} = \{ y \in \mathbb{R} : x = y \} = \{ x \}

    So each point's equivalence class is itself.
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  8. #8
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    Quote Originally Posted by Defunkt View Post
    The equivalence class of x \in \mathbb{R} is [x] = \{ y \in \mathbb{R} : (x,y) \in f \}  = \{ y \in \mathbb{R} : f(x)=y \} = \{ y \in \mathbb{R} : x = y \} = \{ x \}

    So each point's equivalence class is itself.
    Aha, that's what I was hoping to see.

    Beautify! Thanks again.
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