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Math Help - bijection

  1. #1
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    bijection

    It is given that
    (1) f: A --> B is injective iff f has a left inverse. (f has a left inverse if there is a function g: B --> A such that g(f(a)) = a for all a in A)
    (2) f is surjective iff f has a right inverse. (f has a right inverse if there is a function g: B --> A such that f(g(b)) = b for all b in B)

    I need to prove that
    f is a bijection iff there exists g: B --> A such that f(g(b)) = b for every b in B and g(f(a)) = a for every a in A.

    So if I am given (1) and (2) above, can I prove it like this?
    i. Suppose f is bijective. That is, f is injective and surjective. Then by (1) there is a function g: B --> A such that g(f(a)) = a for every a in A since f is injective, and by (2) f(g(b)) = b for every b in B. Thus, g is the inverse of f.
    ii. Suppose f has an inverse g: B --> A such that f(g(b)) = b for every b in B and g(f(a)) = a for every a in A. Then by (1) and (2) f is injective and surjective. Therefore, f is bijective.

    Will this work?
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  2. #2
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    Quote Originally Posted by dori1123 View Post
    It is given that
    (1) f: A --> B is injective iff f has a left inverse. (f has a left inverse if there is a function g: B --> A such that g(f(a)) = a for all a in A)
    Let me do this one and you try the other one.

    Say f: A\to B is injective. Define a function g:B\to A in the following way: let b\in B if b=f(a) for a\in A then define g(b) = a otherwise define g(b) = b. It is important to show this is well-defined meaning it makes sense how we defined this function. Perhaps there are several values of a such that b=f(a) - if so then which one do we choose for g(b) = a ? This is not a problem because the function is injective thus these values of a are unique. This defines a function g: B\to A. Now we need to confirm that g(f(a)) =a. And this is of course true on how we defined g above.

    You try the converse and the second part.
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