# Abstract Algebra: Proof Help

• Aug 31st 2008, 09:18 AM
rubbermagnet
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.
• Aug 31st 2008, 04:59 PM
NonCommAlg
Quote:

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.

$\displaystyle a*b=\frac{ab}{a+b}$ and $\displaystyle b*a=\frac{ba}{b+a}=\frac{ab}{a+b}=a*b.$ so $\displaystyle *$ is commutative. to prove associativity we need to show that $\displaystyle (a*b)*c=a*(b*c).$ we have:

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