Let $A=\{x^2:0 I have been trying to solve the problem. Could someone point me in the right direction? Thanks in advance for your time.
One-0ne ,onto function related problem
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elementary-set-theory
functions
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1What 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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1Thanks Stefan Hansen for makiung the image readable. – 2012-12-19
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2@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 rationality – 2012-12-19
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0@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
2 Answers
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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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0Sir, do you mean that '$t$'is taken from $A$ and $7t+1$ is from the set $B$? – 2012-12-19
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0@user33640: Yes. you are right. He noted the right map from $(0,1)\to (1, 8)$. Check it. – 2012-12-19
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0sir,why are you taking elements of the form $(a,b)$ from the sets $A$ and $B$..If you clarify it,i will be grateful. – 2013-01-10
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0These $(a,b)$ 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
$A=(0,1)$ and $B=(1,8)$. – 2013-01-10