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Math Help - Two quick questions regarding Partitions and Equivalence Relations

  1. #1
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    Two quick questions regarding Partitions and Equivalence Relations

    I'm confused on how to do the following problems for my Proofs class (R represents the set of real numbers):

    1. For the following equivalence relation, describe the corresponding partition.
    Let ~ be the relation on R - {0} given by x ~ y iff xy > 0, for all x,y elements of R - {0}

    2. For the following partition, describe the corresponding equivalence relation.
    Let D be the partition of R^2 consisting of all circles in R^2 centered at the origin (the origin is considered a "degenerate" circle).

    For the second problem, my professor started us off by saying we should write, "Let ~ be the equivalence relation on R^2 given by (x,y) ~ (z,w) iff __________."

    My professor gave a few examples but they weren't the same "style" (for lack of a better word) as the homework problems.

    And, just to clarify, R - {0} is the non-zero real numbers, right?
    Last edited by Lprdgecko; December 2nd 2010 at 10:02 PM. Reason: Spelling mistake
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  2. #2
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    On the first problem, I started off by having P = {[x] | x is an element of R - {0}} (where [x] is the equivalence class), and I think the next step is to determine which non-zero real numbers y can be multiplied by a non-zero real number x such that xy < 0 ?
    Last edited by Lprdgecko; December 2nd 2010 at 10:16 PM.
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  3. #3
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    hint: if x>0 then what is the condition on y such that xy>0? And similarliy what if x<0?

    and for 2 - What is the eqiation of a circle centered at (0,0)?
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  4. #4
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    Well, if x>0, then y must also be greater than 0, and if x<0, then y<0 ... That's pretty much what I was thinking, I'm just a little confused on how to write it out as a partition?
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  5. #5
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    It's been a while since I've used equations of circles, but if I remember correctly, isn't a circle centered at the origin x^2 + y^2 = 0 ?
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  6. #6
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    partition 1 - {x|x>0}
    partition 2 - {x|x<0}

    i think it is as simple as that
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  7. #7
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    yes - so what should be relation between (x,y) and (z,w) such that they lie on the same circle?
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  8. #8
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    Quote Originally Posted by aman_cc View Post
    yes - so what should be relation between (x,y) and (z,w) such that they lie on the same circle?
    They would lie on the same circle iff x^2 + y^2 = z^2 + w^2, which implies x=z and y=w. Is that all the problem is asking for?
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  9. #9
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    "x^2 + y^2 = z^2 + w^2, which implies x=z and y=w."

    This is wrong !! Think about it carefully
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  10. #10
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    Quote Originally Posted by aman_cc View Post
    "x^2 + y^2 = z^2 + w^2, which implies x=z and y=w."

    This is wrong !! Think about it carefully
    Originally all I had written down was (x,y) , (z,w) lie on the same circle iff x=z and y=w, but I added in the x^2+y^2 part... is it still wrong?
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  11. #11
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    (x,y)~(z,w) if x^2+y^2 = z^2 + w^2

    1. Now you should check that above is an equivalence relation
    2. Also understand very clearly that x^2+y^2 = z^2 + w^2 does not => that x=z and y=w. And this is the part which is wrong in your argument
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  12. #12
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    Quote Originally Posted by aman_cc View Post
    (x,y)~(z,w) if x^2+y^2 = z^2 + w^2

    1. Now you should check that above is an equivalence relation
    2. Also understand very clearly that x^2+y^2 = z^2 + w^2 does not => that x=z and y=w. And this is the part which is wrong in your argument
    Ah, yes, I see why I made that mistake now. I think I was trying to use the formula of a circle, but also stuck on the idea that x=z and y=w, but that obviously isn't true because, on the unit circle for example, the points (1,0) and (-1,0) are on the same circle yet 1 does not equal -1. ... Thanks for the help!
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  13. #13
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    correct
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