1. Prove injection and surjection

I need to prove injection and surjection of the following function defined as such that where

It's asked to find domain, range (which I've already found), just that I don't know how to prove that is injective and surjective. The idea I had to prove injectivity was to determine another function, for example so that I can get an equality... well... I don't think that's the right track.

2. Originally Posted by Thelastx
I need to prove injection and surjection of the following function defined as such that where

It's asked to find domain, range (which I've already found), just that I don't know how to prove that is injective and surjective. The idea I had to prove injectivity was to determine another function, for example so that I can get an equality... well... I don't think that's the right track.

It might help to ignore the "x" since that is assumed anyway. h maps $$ to $$.

"Injection" means "if h(u)= h(v) then u= v". Okay, suppose h maps <a, b, c> and <x, y, z> to the same thing. Since h(<a, b, c>)= <b, -c, a> and h(<x, y, z>)= <y, -z, x>, in order to have h(<a, b, c>)= h(<x, y, z>) we must have <b, -c, a>= <y, -z, x> which means b= y, -c= -z, a= x. What does that tell you?

"Surjection" means that, for any <a, b, c>, we can find <x, y, z> so that h(<x, y, z>)= <a, b, c>. Now, h(<x, y, z>)= <y, -z, x>. Can you always find x, y, z so that is equal to <a, b, c>?

3. thanks for your help but i don't fully understand your explanation, i dunno what does mean "maps."