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 

For $x\in{(k,k+1)}$ with $k\geq 2$ and $k$ even define $f(x)=\frac{1}{2x-k}$; for $x\in(0,1]$ define $f(x)=\frac{x+1}{2}$; set $f(0)=0$. Now it remains to map $A=\{2,3,4,5,..\}$ bijectively to $\{1/2,1/4,1/6,1/8..,\}$ to do this define $f(x)=\frac{1}{2(x-1)}$  on $A$.  This should give you the desired bijection. Please, let me know if I have made any silly error this time. 
