Binary operations on groups
I am trying to prove that a set with the specified binary operation below is a group? Here is the problem
on R* x R, (a, b) * (c,d) = (ac, bc + d)
I know that I have to prove that it is has
1.) an identity: a * x = a
2.) an inverse: a * x = e
The thing that throws me off is the (ordered pair). How do I start it off?
This is how I started off trying to prove the identity
a*x = a
(a,b) * (x1, x2) = (a,b) ?? DOn't know if I can use x1,x2 ??
(a*x1, b*x1 + x2 ) = (a,b)
then I get two equations:
a *x1 = a --> x1 = 1
b*x1 + x2 = b --> b*1 + x2 = b --> x2 = 0
NOw I plug x1 and x2 back into original eqn:
(a *1, b*1 + 0 ) = (a, b)
(a,b) = (a,b)
haha, I started off not knowing how to go about and the mere fact of coming to this site and trying to ask a question got me going in the right direction and I ended up solving my own problem, (Clapping). My thanks for this website!