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Partition on R^3

  1. Mar 31, 2008 #1
    1. The problem statement, all variables and given/known data
    Suppose that we partition R^3 into horizontal planes. What equivalence relation is associated with this partition? Suppose that we partition R^3 into concentric spheres, centered at (0,0,0). What equivalence relation is associated with this partition?


    2. Relevant equations



    3. The attempt at a solution

    Since it is on R^3, I know that I need to come up with a partition that has an x, y and z coordinate, right?

    Could the equivalence relation be (x,y,z)~(a,b,c) if and only if x^2=a^2?

    For the second one, could it be something like (x,y,z)~(0,0,0) if and only if x^2=a^2

    Thank you very much
     
  2. jcsd
  3. Mar 31, 2008 #2
    In both cases equivalent points lie on the same plane (sphere). What are the equations of these planes (spheres)?
     
  4. Mar 31, 2008 #3

    HallsofIvy

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    There are two questions here. Which you talking about?

    The equation of any horizontal plane is z= z0. Okay, what must be true of (x1, y1, z1) and (x2, y2, z2) in order that they be on the same plane?

    A sphere centered at (0,0,0) has equation x2+ y2+ z2= R2. What must be true of (x1, y1, z1) and (x2, y2, z2) if they lie on the same sphere?
     
  5. Mar 31, 2008 #4
    Thank you very much

    Thank you
     
  6. Apr 1, 2008 #5

    HallsofIvy

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    No, that was the problem before with planes. If (x1, y1, z1) and (x2, y2, z2) lie on the same sphere then they must both satisfy the equation of that sphere: x12+ y12+ z12= R2 and x22+ y22+ z22= R2 so
    x12+ y12+ z12=x22+ y22+ z22.
     
  7. Apr 6, 2008 #6
    Thank you very much

    Regards
     
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