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Math Help - Question on a surjective map

  1. #1
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    Question on a surjective map

    If f:A \rightarrow B is surjective and A is finite, then B is finite. True or false?
    I think this is false but my book says it is true.

    I think its false by counter-example:
    Let A= (-\frac{\pi}{2},\frac{\pi}{2}) and let f(x) = tan(x). Then B is infinite?

    Am I wrong about this?
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  2. #2
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    Re: Question on a surjective map

    But your A is infinite as well despite the assumption.
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  3. #3
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    Re: Question on a surjective map

    Quote Originally Posted by worc3247 View Post
    If f:A \rightarrow B is surjective and A is finite, then B is finite. True or false?
    Easy to prove:

    Since A is finite, let h be a finite enumeration of A.

    Let f be a surjection from A onto B.

    Define a function g from B as follows:

    g(y) = the least n such that f(h(n)) = y.

    So g is an injection from B into a finite set.
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