CSIR JUNE 2011 PART B QUESTION 25 SOLUTION (SUBSCRIBE MY BLOG TO GET REGULAR UPDATES))

Let $A = \{x^2 : 0<x<1\}$ and $B = \{x^3 : 1<x<2\}$. 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-one.

Solution: We observe that $A = [0,1]$ and $B = [1,8]$. This is because of  $f(x) = x^2$ and $g(x) = x^3$ are continuous strictly increasing functions on $[0,1]$ and $[1,2]$ and we have $A = f([0,1])$ and $B = g([1,2])$.

 1. (true) Both $A$ and $B$ are uncountable sets and the function defined by $h(x) = 8x$ defines a bijection between $A$ and $B$.

2. (false) If $x$ is a rational number then $8x$ is also a rational number. Hence the bijection $h(x)$ from $A$ to $B$ (given above) takes rationals to rationals.

3. (false) Let $f$ be a one to one function which is onto. This means that it is a bijection. So option 3 says that there is no bijection between $A$ and $B$. But we have constructed a bijection between $A$ and $B$ in option (1).

4. (false) Similar to (3).

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