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Math Help - Prove injection and surjection

  1. #1
    Newbie Thelastx's Avatar
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    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.

    Thanks in advance.
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  2. #2
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    Quote Originally Posted by Thelastx View Post
    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.

    Thanks in advance.
    It might help to ignore the "x" since that is assumed anyway. h maps <a_0, a_1, a_2> to <a_1, -a_2, a_0>.

    "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>?
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  3. #3
    Newbie Thelastx's Avatar
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    thanks for your help but i don't fully understand your explanation, i dunno what does mean "maps."
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