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Math Help - Equivalence relation?

  1. #1
    Junior Member Fnus's Avatar
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    Equivalence relation?

    I've added a picture with the information about the relation.

    Reflexive: I'd say it's reflexive since x*x would stiil belong to the rational numbers, right?
    But is it enough to just write that, or do I need to do something more?

    Symmetric: Again I'd say it is symmetric because multiplication is commutative.
    Is that enough? xD

    Transitive: No idea, I'm so bad at transitive, so any help would be appreciated, thanks!
    Attached Thumbnails Attached Thumbnails Equivalence relation?-math.bmp  
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  2. #2
    Member Haven's Avatar
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    Your idea for the reflexivity of the relation is good

     xTx = \{x\in\\Q ^ {x^2}\in\\Q\}

    which is obvious if we let  x = a/b\rightarrow\\{x^2} = {a^2}/{b^2} . which will be in Q

    Symmetric:

     xTy = \{x\in\\Q ^ {x*y}\in\\Q\}\rightarrow\{y\in\\Q ^ {y*x}\in\\Q\} = yTx

    since multiplication in Q is communative, i.e.,  xy = yx .
    If  x = a/b and  xy = c/d , since both x and xy are rational then it follows there must be a y in Q such that  xy = {a/b}*{y}= c/d . Apply the same argument but for x and you show  xTy = yTx

    Transitive:

    if  xTy and  yTz then  xTz

    by  xTy , we know x and xy are in Q, and by  yTz we know y and yz are in Q. So we want to know if xz is in Q. It follows if y and yz are in Q then z must be in Q, by the same argument for the Symmetric. So if x is in Q and z is in Q, then xz must be in Q.
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