Define a binary operation on the rationals so that the map defined by is an isomorphism between sets with binary operations. Prove that is an isomorphism.

THIS IS WHAT I HAVE TO DO

"We now give an outline to show two binary structures (S,*) and (S,*') are isomorphic.

To prove it is an isomorphism i have to:

1) Define the function that gives the isomorphism of S with S'. Now this means that we have to describe, in some fashion, what is for every s S.

2) Show that is a one-to-one function. That is, suppose that in S' and deduce from this that x = y in S.

3) Show that is onto S'. that is, suppose that s' S' is given and show that there does exist s S such that

4) Show that for all x,y S. This is just a question of computation. Compute both sides of the equation and see whether they are the same."

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I have defined = u + v +

1) The function has already been defined

2) Lets Suppose

then 7x + 1 = 7y + 1

7x = 7y

x = y

Therefore it is one-to-one

3) I dont even know were to start

4)

7(u v) + 1 = 7u + 1 + 7v + 1

7(u v)= 7(u + v + )

u v= u + v +

So what i need help is part 3) and i would like some1 to tell me if i had done the other parts correctly?