## Specify if the operation is a group

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### Specify if the operation is a group

The symbol RxR represents the set of all ordered pairs (x,y) of real numbers. RxR may therefore be identified with the set of all the points in the plane. Which of the following subsets of RxR, with the indicated operation is a group? Which is an abelian group.

A group is a set G with an operation * which satisfies the axioms
1.) * is associative.
2.) There is an element e in G such that a*e=a and e*a=a for every element a in G.(identity)
3.) For every element a in G, there is an element a' in G such that a*a'=e and a'*a=e (inverse)

Question 1.)
(a,b)*(c,d)=(ac,bc+d) on the set {(x,y) in RxR: with x not equal to 0}

So to find out if this operation is a group, we must find out if it is associative, has an identity, and has an inverse.
I have found that it is associative, has an identity, and has an inverse.

NOW the real question is this:
Is (a,b)*(c,d)=(ac,bc+d) a group on the set RxR, including 0 ?

Does anything even change including 0? i don't think it does. Anybody have any suggestions or examples which proves it is not a group?
Spaghett
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