G is the set of the real numbers with the operation xyxy1 Fi

G is the set of the real numbers with the operation x*y=x+y+1. Find an isomorphism f:R->G and show that it is an isomorphism.

I will prove G is group

1. will show (x* y)* z = x*(y* z)

(x* y)* z = (x*y)+ z+ 1

= x+ y+ z+ 2

x*(y* z) = x+(y*z)+1

= x+ y+ z+ 2 \\ (x* y)* z = x*(y* z)

Then G is associative

2. from -1Î G so

x* -1 = x+(-1) +1 = x

-1*x = -1 +x-1 = x

-1 is identity in G

3.Let xÎ G

and -(x+2) Î G

x* -(x+2) = x- (x+2)+1 = -1

-(x+2)* x = -(x+2)+x+1 = -1

x* -(x+2) = -1 = -(x+2)* x

Then -(x+2) is inverse 0f x in G

G is group

Let f : R® G defined by f (x) = x-1

2. we will show f is onto

let yÎ G choose x=y+1 Î R

y+1 = x

so f (x) = x-1 =(y+1)-1 = y

f is onto

3. will show f is homomorphism

f (xy) = xy -1

f (x) * f (y) = (x-1) *(y-1)

= (x-1)+(y-1)+(x-1)(y-1)

= x-1+ y-1+ xy-y-x+1

= xy-1

= f (xy)

f is homomorphism

so it is isomorhism