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.

1. (

2. (

3. (

4. (

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).**Share to your groups:**

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