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Let $f:X\rightarrow X$ be defined by $f(x)=x^2$, where $X\subset \mathbb{C}$. What is an example of $X$ so that $f$ is bijective? Neither injective nor surjective? Surjective but not injective? Injective but not surjective?

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That sounds a lot like homework. Can it really be that you have made no progress at all on any of the subquestions? –  Henning Makholm Oct 21 '11 at 2:04
    
What have you tried so far? Any guesses? –  Zev Chonoles Oct 21 '11 at 2:05
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@smanoos SE accepts questions of all difficulties. When it is clear the question is homework, it is encouraged to give hints rather than answers and ask the OP what s/he has already tried (in order to better diagnose the difficulty). –  Austin Mohr Oct 21 '11 at 2:22
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@johnny, setting $X=i^2$ makes no sense here. $i^2$ is just a number (namely $-1$), but $X$ is supposed to be a set of numbers -- specifically, a subset of $\mathbb C$. As for fitting complex numbers into $x^2$, surely you have learned how to multiply complex numbers? $x^2$ is just a complex number multiplied by a complex number that happens to be itself. –  Henning Makholm Oct 21 '11 at 2:33
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Now, to be a bit concrete, one subset of $\mathbb C$ you could try to use as $X$ would be $\mathbb R$. Which, if any, of the four cases, does that fit into? –  Henning Makholm Oct 21 '11 at 2:50

1 Answer 1

Suppose you set $X=\mathbb{R}^+$, the set of positive real numbers. Then you have the function being bijective. If $X=\mathbb{R}$, then the function is neither injective nor surjective. Note that $\mathbb{R}\subset \mathbb{C}$, so this makes sense. Use similar argument to obtain the rest.

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