elementary set theory - How to construct a one-to one correspondence between$left [ 0,1 right ]bigcup left [ 2,3 right ]bigcup ..$ and $left [ 0,1 right ]$

How can I construct a one-to one correspondence between the Set $\left [ 0,1 \right ]\bigcup \left [ 2,3 \right ]\bigcup\left [ 4,5 \right ] ...$ and the set $\left [ 0,1 \right ]$ I know that they have the same cardinality

Answer

Suppose that you had a bijection $f:[0,1]\to(0,1]$. Then you could decompose $[0,1]$ as

$$\begin{align*} [0,1]&=\left[0,\frac12\right]\cup\left(\frac34,1\right]\cup\left(\frac58,\frac34\right]\cup\left(\frac9{16},\frac58\right]\cup\dots\\ &=\left[0,\frac12\right]\cup\bigcup_{n\ge 1}\left(\frac{2^n-1}{2^{n+1}},\frac{2^{n-1}+1}{2^n}\right]\;, \end{align*}$$

map $[0,1]$ to $\left[0,\frac12\right]$ in the obvious way, and for $n\ge 1$ map $[2n,2n+1]$ to $\left(\frac{2^n-1}{2^{n+1}},\frac{2^{n-1}+1}{2^n}\right]$ using straightforward modifications of $f$ for each ‘piece’. I’ll leave that part to you unless you get stuck and ask me to expand; the hard part is finding $f$. Here’s one way:

$$f:[0,1]\to(0,1]:x\mapsto\begin{cases} \frac12,&\text{if }x=0\\\\ \frac1{2^{n+1}},&\text{if }x=\frac1{2^n}\text{ for some }n\ge 1\\\\ x,&\text{otherwise}\;. \end{cases}$$

In other words, $f$ is the identity map except on the set $\displaystyle{\{0\}\cup\left\{\frac1{2^n}:n\ge 1\right\}}$, which it shifts one place ‘forward’ like this:

$$0\overset{f}\mapsto\frac12\overset{f}\mapsto\frac14\overset{f}\mapsto\frac18\overset{f}\mapsto\frac1{16}\overset{f}\mapsto\dots\;.$$