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Math Help - Set Theory

  1. #1
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    Set Theory

    Hello, I am doing my homework to study for my exam but I am stuck on this problem...plz help me
    Thank you in advance.

    Let R denote the real numbers, and let P be the binary relation on R X (R - {0} ) such that (x,y) P (z,w) if and only if xw = yz. Prove that P is an equivalence relation, and show that every equivalence class contains a unique element (or representative) of the form (r,1).
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  2. #2
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    Quote Originally Posted by jenjen View Post
    Hello, I am doing my homework to study for my exam but I am stuck on this problem...plz help me
    Thank you in advance.

    Let R denote the real numbers, and let P be the binary relation on R X (R - {0} ) such that (x,y) P (z,w) if and only if xw = yz. Prove that P is an equivalence relation, and show that every equivalence class contains a unique element (or representative) of the form (r,1).
    You need to check that P satisfies the conditions in the definition of an
    equivalence relation:

    1. Reflexivity: aPa
    2. Symmetry: if aPb then bPa
    3. Transitivity: if aPb and bPc then aPc.

    a. Let a=(x,y), y!=0, then xy = yx so aPa, and condition 1 is satisfied.

    b. Let a=(x,y) b=(u,v), y, v !=0, and aPb. Therefore xv=yu so yu = xv,
    and as ordinary multiplication is comutative uy=vx, and so bPa and
    condition 2 is satisfied.

    c. Let a=(x,y), b=(u,v), c=(z,w), y, v, w !=0, and aPb and bPc.
    Then we have xv = yu and uw = vz. and as y!=0 we can write

    u = xv/y,

    so:

    (xv/y)w = vz,

    hence:

    xw = yz

    which is aPc, so condition 3 is satisfied and hence P is an equivalence
    relation on R X (R - {0} ).

    For the second part observe that if a=(x,y) is in R X (R - {0} ), then
    aPb, where b=(x/y,1). So every element of the set is in an equivalence
    class with an element of the form (r,1). Now suppose there are two distinct
    elements of this form in the same equivalence class say (r1,1) and (r2,1),
    with r1 != r2, then:

    r1.1 = 1.r2

    so we are forced to conclude that r1=r2 which is a contradiction.

    hence we have proven that every element in the set is eqi=uivalent to a
    unique element of the form (r,1) which may be taken to be representative of
    the equivalence class.

    RonL
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