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Math Help - Abstract Algebra: Proof Help

  1. #1
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    Abstract Algebra: Proof Help

    I need help setting up this problem.

    Let Q be the set of positive rationals. Consider the operation * defined by a*b =ab/a+b. Prove that * is an associative, commutative binary operation on Q. I need to make sure i clearly identify and state places in the proof where you use properties of the operations + and . on Q.
    I know what I need to do, I just do not know how to set up the proof. If I need to list what I have then I can. It would be great to get some feed back.
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  2. #2
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    Quote Originally Posted by rubbermagnet View Post
    I need help setting up this problem.

    Let Q be the set of positive rationals. Consider the operation * defined by a*b =ab/a+b. Prove that * is an associative, commutative binary operation on Q. I need to make sure i clearly identify and state places in the proof where you use properties of the operations + and . on Q.
    I know what I need to do, I just do not know how to set up the proof. If I need to list what I have then I can. It would be great to get some feed back.
    a*b=\frac{ab}{a+b} and b*a=\frac{ba}{b+a}=\frac{ab}{a+b}=a*b. so * is commutative. to prove associativity we need to show that (a*b)*c=a*(b*c). we have:

    (a*b)*c=\frac{(a*b)c}{(a*b)+c}=\frac{(\frac{ab}{a+  b})c}{\frac{ab}{a+b} +c}=\frac{abc}{ab+ac+bc}. and: a*(b*c)=\frac{a(b*c)}{a+(b*c)}=\frac{a (\frac{bc}{b+c})}{a+\frac{bc}{b+c}}=\frac{abc}{ab+  ac+bc}. therefore: (a*b)*c=a*(b*c).
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