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Injection, surjection, and bijection

  1. Oct 13, 2009 #1
    I'm having trouble understanding just what is the difference between the three types of maps: injective, surjective, and bijective maps. I understand it has something to do with the values, for example if we have T(x): X -> Y, that the values in X are all in Y or that some of them are in Y...
    Honestly I'm just incredibly confused about the terms. If someone could give me a straightforward way of explaining each of them I would very much appreciate it.
  2. jcsd
  3. Oct 13, 2009 #2

    Ben Niehoff

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  4. Oct 13, 2009 #3
    Remember the definition of a function f : X --> Y. It must satisfy two essential conditions:

    1. Every element of X gets mapped to something in Y.
    2. That something in Y is unique for each element of X.

    Injections and surjections are special kinds of functions that also have one of these properties going in the other direction:

    (Surj.) Every element of Y is mapped to by some element of X.
    (Inj.) The element of X that maps to a particular value in Y is unique.

    A function which is both surjective and injective is called bijective.
  5. Oct 13, 2009 #4
    Wow, thank you so much! That was exactly the explanation I was looking for.
    This will make my linear class so much easier to follow
  6. Oct 13, 2009 #5
    Glad to have been of help. :)
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