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Math Help - Partitions

  1. #1
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    Partitions

    Let R2 denote the usual (x, y)-plane. We can define a partition of R2 by using the
    lines with slope 2. Describe the corresponding equivalence relation ∼ by giving the
    conditions on the coordinates so that (x1, y1) ∼ (x2, y2).
    I have we must have (y2-y1)/(x2-x1)=2
    Then y2-y1=2x2-2x1
    y2-2x2=y1-2x1
    So x1=2x2?
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  2. #2
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    Quote Originally Posted by kathrynmath View Post
    Let R2 denote the usual (x, y)-plane. We can define a partition of R2 by using the
    lines with slope 2. Describe the corresponding equivalence relation ∼ by giving the
    conditions on the coordinates so that (x1, y1) ∼ (x2, y2).
    I have we must have (y2-y1)/(x2-x1)=2
    Then y2-y1=2x2-2x1
    y2-2x2=y1-2x1
    So x1=2x2?
    Your basic algebra is off.
    \dfrac{y_2-y_1}{x_2-x_1}=2 means  y_2-y_1 = 2x_2-2x_1 .
    Nothing more can be done. You can rearrange it, but not combine terms.
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  3. #3
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    Okay, your equivalence classes are the individual lines. Two points are equivalent, (x_1, y_1)~(x_2, y_2) if and only if they lie on the same line. That is, as you say, \frac{y_2- y_1}{x_2- x_1}= 2 which is the same as saying that y_2- y_1= 2x_2- 2x_1 and the same as y_2- 2x_2= y_1- 2x_1. But I cannot see how you got " x_1= 2x_2" from that! Any one of those first three equations will do. The simplest, in my opinion, is the first: (x_1, y_1)~ (x_2, y_2) if and only if \frac{y_2- y_1}{x_2- x_1}= 2.
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  4. #4
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    So that's all I would have to say for describing the equivalence relating by saying they are only related if (y2-y1)/(x2-x1)=2?
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  5. #5
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    Yes.
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