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Math Help - Examples of Equivalence Relations.

  1. #1
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    Examples of Equivalence Relations.


    I was working on #6. Once i find it is an equivalence relation how do i describe the partition?
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    Re: Examples of Equivalence Relations.

    Quote Originally Posted by Aquameatwad View Post
    I was working on #6.
    What does \mathcal{F}(\mathbb{R}) denote?

    Quote Originally Posted by Aquameatwad View Post
    Once i find it is an equivalence relation how do i describe the partition?
    By considering a lot of examples. Fix some f(x) and describe all functions equivalent to it, as well as functions not equivalent to it. What information needs to be provided to identify an equivalence class?
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    Re: Examples of Equivalence Relations.

    I think F(R) is set of all functions of real numbers. is it where f(x)=g(x) for some x in R?

    To identify an equivalence class? well the equiv relation is f(0)=g(0), but that's all we know, so for every [f(0)]: f(0)=g(0) ...
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    Re: Examples of Equivalence Relations.

    Quote Originally Posted by Aquameatwad View Post
    I think F(R) is set of all functions of real numbers. is it where f(x)=g(x) for some x in R?
    To identify an equivalence class?
    If \mathcal{F}\left( \mathbb{R} \right) is the set of all real valued functions then every real number determines an equivalence class. WHY?
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    Re: Examples of Equivalence Relations.

    because every function f(x) is equal to another function g(x)? so f(x) = g(x) for every real number x?
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  6. #6
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    Re: Examples of Equivalence Relations.

    Quote Originally Posted by Aquameatwad View Post
    because every function f(x) is equal to another function g(x)? so f(x) = g(x) for every real number x?
    Are you looking at #6?
    As I read it f\sim g if and only if f(0)=g(0). Has nothing to do with any x.
    If c\in\mathbb{R} then define [c]=\{f:f(0)=c\}.
    Can you show that the collection \left\{ {[c]:c \in \mathbb{R}} \right\} partitions \mathcal{F}(\mathbb{R})~.
    Last edited by Plato; March 21st 2012 at 03:19 PM.
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    Re: Examples of Equivalence Relations.

    Yes, can you explain without using math talk. All i see are lines and squiggles.
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    Re: Examples of Equivalence Relations.

    I would still consider examples.
    Quote Originally Posted by emakarov View Post
    Fix some f(x) and describe all functions equivalent to it
    Quote Originally Posted by Aquameatwad View Post
    is it where f(x)=g(x) for some x in R?
    As Plato said, no. Let, e.g., f(x) = x^2 + 1. How would you describe all functions equivalent to it? Non-equivalent to it?
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    Re: Examples of Equivalence Relations.

    Well there's the 'x' again. i thought plato said not to use an x. So now im getting even more confused

    Anyway to describe the functions equivalent to f(x) is for some function g(x), f(x)=g(x) to be equivalent right?
    and not equivalent f(x)≠g(x) right?

    So what does this have to do with describing the partition?
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    Re: Examples of Equivalence Relations.

    Quote Originally Posted by Aquameatwad View Post
    Well there's the 'x' again. i thought plato said not to use an x.
    Plato said that the definition of f ~ g in problem 6 does not use things like "for some x" or "for all x." In particular, f ~ g does not mean "f(x) = g(x) for every real number x," which is a quote from post #5.

    Quote Originally Posted by Aquameatwad View Post
    Anyway to describe the functions equivalent to f(x) is for some function g(x), f(x)=g(x) to be equivalent right?
    This is not the definition of f ~ g from problem 6.

    So, the question remains: describe all function g(x) such that g(x) ~ x^2 + 1.
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  11. #11
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    Re: Examples of Equivalence Relations.

    how do we decide if f and g are equivalent? we compare f(0) and g(0). for example, if f(x) = x^2 + 1, then f(0) = 0^2 + 1 = 1, so if g ~ f, g(0) = 1. that's all we need to know, and in fact, it's all we CAN say about such functions g. so one way of characterizing [f] is saying "all g with the same y-intercept as f". how many different y-intercepts are there (that is, how many points do we have on the y-axis)?
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