Thread: I Need a Binary Operation

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.

By Isomorphism i mean i need to find a binary operation such that for all u, v in ,
$\displaystyle \phi(u \bullet v) = \phi(u) + \phi(v)$

Originally Posted by kbartlett

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.

By Isomorphism i mean i need to find a binary operation such that for all u, v in ,
$\displaystyle \phi(u \bullet v) = \phi(u) + \phi(v)$

You want $\displaystyle \phi (u\bullet v) = \phi (u) + \phi (v)$ to hold for all $\displaystyle u,v$.
Therefore, $\displaystyle 7(u\bullet v) + 1 = (7u + 1) + (7v+1)$.
Thus, $\displaystyle u\bullet v = u + v + \tfrac{1}{7}$

thankyou, it seems so obvious now. The big words just confussed me .

Do u know how i could go about Proving that $\displaystyle \phi$ is an isomorphism.