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Math Help - 1:1 and onto

  1. #1
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    1:1 and onto

    Is it possible for a function to be 1:1 and not onto and is it possible for a function to be onto but not 1:1? The catch is the function is from a set A->A. I can't find a set A and a function f so f:A->A that fits these (two separate functions and sets to meet each condition). Any hints on this are greatly appreciated.
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  2. #2
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    Yes, it is possible.
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  3. #3
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    Quote Originally Posted by vassago View Post
    Is it possible for a function to be 1:1 and not onto and is it possible for a function to be onto but not 1:1? The catch is the function is from a set A->A. I can't find a set A and a function f so f:A->A that fits these (two separate functions and sets to meet each condition). Any hints on this are greatly appreciated.
    Is it possible for a function to be 1:1 and not onto and is it possible for a function to be onto but not 1:1?
    <br />
f: \mathbb{Z} \rightarrow \mathbb{Z} by x \mapsto 2x. 1-1 not onto

    a function to be onto but not 1:1?
    f: \mathbb{Z} \rightarrow \{0\} by x \mapsto 0. onto not 1-1
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  4. #4
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    Quote Originally Posted by GaloisTheory1 View Post
    <br />
f: \mathbb{Z} \rightarrow \mathbb{Z} by x \mapsto 2x. 1-1 not onto

    f: \mathbb{Z} \rightarrow \{0\} by x \mapsto 0. onto not 1-1
    Of, if you must have functions that map a set onto itself, f: \mathbb{Z} \rightarrow \mathbb{Z}[/tex], f(x)= x if x\le 0, f(x)= x-1 for x> 0. That is onto but not 1-1 since f(0)= f(1)= 0.

    If A were a finite set neither of these would be possible.
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