I Need a Binary Operation

**kbartlett** · Oct 8th 2008, 07:21 AM

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

,
$\phi(u \bullet v) = \phi(u) + \phi(v)$

**ThePerfectHacker** · Oct 8th 2008, 07:33 AM

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

,
$\phi(u \bullet v) = \phi(u) + \phi(v)$

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

**kbartlett** · Oct 8th 2008, 09:14 AM

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

.

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

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