What do you mean by 'same'? If f is an isomorphism then f(xy)=f(x)f(y), if that's what you mean.
Is there a missing 1 there?
Where is 0? Why haven't you written down x or x^x+x? (You have, by the way, but you have made a strange choice of representatives of the elements, which is why I ask, since it implies you've not really understood what you're doing.