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Math Help - Equivalence Classes and Partial Orders

  1. #1
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    Equivalence Classes and Partial Orders

    I need help with the following:

    f(x) = x^2 +2x + 5 and let E = {-5,-4,-3,-2,-1,0,1,2,3,4,5}

    Define the relation R in E as follows: xRy if f(x) = f(y).

    I need to find the equivalence classes of R and I'm completely confused, all the examples we've done in class were a lot more simple so I can't get anything out of my notes.


    The second problem I'm pretty much done with:

    For the following 2 relations on the set L of all living people I'm supposed to state if it is reflexive, symm, anti-symm, and transitive. Then if its a partial order indicate if the set has a maximum, minimum, maximals and minimals and explain what they mean. I'm just having trouble with the latter part.

    the relations are:

    xRy if X and Y have the same first name
    xRy if x and y have the same major


    Really appreciate help from anyone. Thanks
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  2. #2
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    Here are hints on the first one.
    \begin{array}{*{20}c}    x &\vline &  { - 5} & { - 4} & { - 3} & { - 2} & { - 1} & 0 & 1 & 2 & 3 & 4 & 5  \\ \hline {f(x)} &\vline &  {20} & {13} & 8 & 5 & 4 & 5 & 8 & {13} & {20} & {29} & {40}  \\ \end{array}
    You can see that \{-5,3\} is one equivalnce class because f(-5)=f(3).
    But \{4\} is also a single class.WHY?
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  3. #3
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    Quote Originally Posted by Plato View Post
    Here are hints on the first one.
    \begin{array}{*{20}c}    x &\vline &  { - 5} & { - 4} & { - 3} & { - 2} & { - 1} & 0 & 1 & 2 & 3 & 4 & 5  \\ \hline {f(x)} &\vline &  {20} & {13} & 8 & 5 & 4 & 5 & 8 & {13} & {20} & {29} & {40}  \\ \end{array}
    You can see that \{-5,3\} is one equivalnce class because f(-5)=f(3).
    But \{4\} is also a single class.WHY?
    So the eq classes are {-5,3}, {-4,2}, {-3,1},{-2,0}. As for {-1}, {4}, and {5}, are they also eq. classes? Despite that the other value isn't in the set?

    Thanks
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  4. #4
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    Quote Originally Posted by plato View Post
    but \{4\} is also a single class.why?
    4R4
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  5. #5
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    Quote Originally Posted by CrShNbRn View Post
    As for {-1}, {4}, and {5}, are they also eq. classes? Despite that the other value isn't in the set?

    Thanks
    Every member of the set E must belong to an equivalence class.
    Last edited by novice; April 8th 2010 at 05:07 PM. Reason: spelling
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