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Let $A=\{x^2:0 and $B=\{x^3:1. Which of the following statements is true?

  1. There is a one to one, onto function from $A$ to $B$.
  2. There is no one to one, onto function from $A$ to $B$ taking rationals to rationals.
  3. There is no one to one function from $A$ to $B$ which is onto.
  4. There is no onto function from $A$ to $B$ which is one to one.

I have been trying to solve the problem. Could someone point me in the right direction? Thanks in advance for your time.

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    What have you tried? A bijection for (1) is readily written down, thus (3) and (4) are falsified immediately. And it's hard to believe that an example you quickly find for (1) is not also a counterexample for (2).2012-12-19
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    Thanks Stefan Hansen for makiung the image readable.2012-12-19
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    @HagenvonEitzen Actually, it's not that hard to believe. I think the first example you're "expected" to find (and the one that leads you into the trap here) is the map that takes $x^2$ to $(x+1)^3$, which is a bijection that doesn't preserve rationality2012-12-19
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    @Zimul8r Oh, I wasn't even able to fall for that. Then again, all we need is that $A,B$ are uncountable and $A\cap \mathbb Q,B\cap \mathbb Q$ are infinite.2012-12-19

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