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Math Help - let f:A ---> B and g:B ---> C (if both are bijective, prove g o f is bijective)

  1. #1
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    let f:A ---> B and g:B ---> C (if both are bijective, prove g o f is bijective)

    Let f: A ---> B and g: B ---> C, if both f, g are bijective, show (prove) that g o f : A ---> C is bijective.
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  2. #2
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    Hi

    There are various ways to prove that assertion, which seems very natural.

    You want g\circ f to be injective and surjective, assuming that f and g have such properties.

    Take two distinct elements a_1 and a_2 in A. f injective \Rightarrow f(a_1)\neq f(a_2). Do you see how to the same idea to prove that g\circ f is injective too.

    f\ \text{surjective}\ \Leftrightarrow\ \text{Im}f = B . g\ \text{surjective}\ \Leftrightarrow\ \text{Im}g=C. (just the definitions of f and g surjectivity). Given a c\in C, why is there a a\in A such that g(f(a))=c ?
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