Abstract Algebra: Proof Help

**rubbermagnet** · August 31st 2008, 09:18 AM

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.

**NonCommAlg** · August 31st 2008, 04:59 PM

Originally Posted by rubbermagnet

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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