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Math Help - surj, bijec, invertible

  1. #1
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    surj, bijec, invertible

    can some one explain me the exact meaning of surjective, bijective, invertible? and the difference hence among them? as i was asked if monotone implies that the function is invertible, said no because it needs to be continuous too, but the prof said that monotone implies invertible as we are not considering surjectve(i think he said this...not sure)but i couldn't catch the meaning...
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  2. #2
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    Quote Originally Posted by Aglaia View Post
    can some one explain me the exact meaning of surjective, bijective, invertible?
    Let f:X-->Y be a function.
    If f(x_1)=f(x_2) then, x_1=x_2 the function is said to be injective (Bourbaki's terminology). Alternate form is one-to-one

    If for any y in Y we can find an x in X such that f(x)=y the function is said to be surjective (Bourbaki's terminology). Alternate form is onto

    If f is both injective and surjective it is bijective.
    It is not difficult to show a function has an inverse function (invertible) if and only if it is bijective.

    There is a useful theorem (Cantor–Bernstein–Schroeder theorem)* that
    if an injective funtion exists between X to Y and Y to X. Then there exists a bijection between the sets.

    *)It was originally proposed by Cantor and proved using Axiom of Choice. Later Bernstein, Schroeder proved it was not necessary.
    as i was asked if monotone implies that the function is invertible, said no because it needs to be continuous too
    Right because for a real function to be invertible it needs to be surjective, in a sense it needs to satisfy the Intermediate Value Theorem. Now if a function is continous then it does satisfy IVT and you have an invertible function when it is striclty monotone (because it gives a unique value for each distinct x). That is a sufficient condition but not necessary (not able to come up with an example though).
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