To determine the existence of a map from a set to another set.

Question

Let $A = \{ x^2| 0<x<1\}$ and $B = \{x^3| 1<x<2\}$. Does there exist a one to one and onto map from A to B?

My attempt:- For a one to one and onto map (basically a bijection) to exist between two sets, their cardinalities must be same. But I do not know how to proceed. Any help will be much appreciated.

Answer 1

As it can be seen $ 0< x <1 \Rightarrow 0 < x^2 < 1 $

$0 < x^2 < 1 \Rightarrow A=(0,1)$

$1<x<2 \Rightarrow 1 < x^3< 8$

$1 < x^3< 8 \Rightarrow B=(1,8)$

let $f(x)=7x+1$

$f(a)=f(b) \Rightarrow 7a=7b \Rightarrow a=b$

let $n \in (1,8)$

$n=f(x)=7x+1 \Rightarrow x=\frac{n-1}{7}$

so $f(x)$ is a bijection between $A$ and $B$