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Math Help - Equivalence Relation Help!

  1. #1
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    Post Equivalence Relation Help!

    I need help in equivalence relationship...

    Question> For points (x1, y1) and (x2, y2) in a plane with rectangular
    coordinate system, let (x1, y1) ~ (x2, y2) mean that either x1=x2 or y1=y2.
    i) Explain why ~ is not an equivalence relation on the set of points in the
    plane.
    ii) And describe the equivalence classes geometrically..And give a complete
    set of equivalence class representatives.

    Thank you!
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  2. #2
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    Quote Originally Posted by Vedicmaths View Post
    I need help in equivalence relationship...

    Question> For points (x1, y1) and (x2, y2) in a plane with rectangular
    coordinate system, let (x1, y1) ~ (x2, y2) mean that either x1=x2 or y1=y2.
    i) Explain why ~ is not an equivalence relation on the set of points in the
    plane.
    ii) And describe the equivalence classes geometrically..And give a complete
    set of equivalence class representatives.
    \left( {1,2} \right) \sim \left( {1, - 2} \right)\,\& \,\left( {1, - 2} \right) \sim \left( { - 1, - 2} \right)\,but\,\left( {1,2} \right)\mathop  \sim \limits^? \left( { - 1, - 2} \right)

    For ii) it is not equivalence relation. So I don't understand the point.
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  3. #3
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Vedicmaths View Post
    I need help in equivalence relationship...

    Question> For points (x1, y1) and (x2, y2) in a plane with rectangular
    coordinate system, let (x1, y1) ~ (x2, y2) mean that either x1=x2 or y1=y2.
    i) Explain why ~ is not an equivalence relation on the set of points in the
    plane.
    ii) And describe the equivalence classes geometrically..And give a complete
    set of equivalence class representatives.

    Thank you!
    Quote Originally Posted by Plato View Post
    \left( {1,2} \right) \sim \left( {1, - 2} \right)\,\& \,\left( {1, - 2} \right) \sim \left( { - 1, - 2} \right)\,but\,\left( {1,2} \right)\mathop  \sim \limits^? \left( { - 1, - 2} \right)
    Perhaps the "or" condition is supposed to be "at least one of x_1 = x_2 or y_1 = y_2? But then the first part is faulty because that is an equivalence relation!

    -Dan
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  4. #4
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    Quote Originally Posted by topsquark View Post
    Perhaps the "or" condition is supposed to be "at least one of x_1 = x_2 or y_1 = y_2?
    No. At least one is equivalent to or.
    In fact, the counter example to transitivity works for that.

    There is a classical equivalence relation similar to that one.
    \left( {a,b} \right) \sim \left( {x,y} \right)\,iff\,\left| a \right| = \left| x \right| \wedge \left| b \right| = \left| y \right|.
    Note that that is and.
    Last edited by Plato; July 7th 2008 at 02:09 PM.
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  5. #5
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    Yes Sir! you are right, I forgot to mention a point in the (ii) one..
    Let (x1, y1) ~ (x2, y2) means that y1=y2. I mistakenly mentioned either and or.
    ii) Describe the equivalence classes geometrically..And give a complete
    set of equivalence class representatives.
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  6. #6
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    Quote Originally Posted by Vedicmaths View Post
    Let (x1, y1) ~ (x2, y2) means that y1=y2.
    ii) Describe the equivalence classes geometrically. And give a complete
    set of equivalence class representatives.
    Well that is much better. Clearly that is an equivalence relation.
    We can describe an equivalence class as “the set of pairs having the same second term”.
    Geometrically that is the set of all horizontal lines. Each class is one of those lines.
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