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Math Help - Equivalence relation

  1. #1
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    Equivalence relation

    Determine whether the following relation is an equivalence relation:

    1. For all x,y element of Real # xRy <-> x-y is an integer.

    2. For all a,b elemnt of Natural # aRb <-> a ends in the same digit in which be ends


    With equivalence relation, i am so lost. I could really use some help here please. This section of the book is just not making much sense to me. Thank you
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by smoothi963 View Post
    Determine whether the following relation is an equivalence relation:

    1. For all x,y element of Real # xRy <-> x-y is an integer.

    2. For all a,b elemnt of Natural # aRb <-> a ends in the same digit in which be ends


    With equivalence relation, i am so lost. I could really use some help here please. This section of the book is just not making much sense to me. Thank you
    For a relation to be an equivalence relation on X, for any a, b and c in X the
    following must hold:

    (i) Reflexivity: a ~ a
    (ii) Symmetry: if a ~ b then b ~ a
    (iii) Transitivity: if a ~ b and b ~ c then a ~ c.

    1) X = R, let a, b and c be in R.

    Clearly aRa <-> a-a is an integer but a-a=0 which is an integer so (i) holds

    Also if a-b is an integer then so is b-a, so (ii) holds

    Now if a-b is an integer, and so is b-c then:

    a-c = (a-b) + (b-c)

    is the sum of two integers and so is an iteger so aRc, so (iii) holds

    Hence we conclude that R is an equivalence relation on R.

    2) Same as above just go through checking reflexivity, symmetry and
    transitivity if the hold the R is an equivelence relation if not it is not

    RonL
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  3. #3
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    Thank you captainblack. I was wondering from

    (i) Reflexivity: a ~ a
    what does a ~ a mean? I don't know what the ~ represent. I am jsut trying to understand this so i can do well on my test. Thank you
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by smoothi963 View Post
    Thank you captainblack. I was wondering from



    what does a ~ a mean? I don't know what the ~ represent. I am jsut trying to understand this so i can do well on my test. Thank you
    In your notation aRa

    RonL
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