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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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    @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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A hint: The sets $A$ and $B$ are described in a somewhat cumbersome way. Find really simple descriptions of these two sets.

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Allow me to point you into the direction of the function $t\mapsto 7t+1$.

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    These $(a,b)$ in Babak's comment are not to be interpreted as elements (ordered pair with components $a$ and $b$) but as open intervals (i.e. (a,b)=\{x\in\mathbb R\mid a). Note that $A=(0,1)$ and $B=(1,8)$.2013-01-10