Find bijection between two sets to show equal cardinalities

[mar_f]

Can I have one more question, please?
I have an example here: [MATH]|[0,1)| = |[0,1) \times N| [/MATH]Can you please help me how can I find bijection between these two sets to show that they have equal cardinalities?
Many thanks.

 

[Steven G]

What have you tried? Even if you know what you did is wrong then please show it to us so we can see where you need help. Thanks.

 

[mar_f]

Hello, the only thing that came to my mind is that I can divide [MATH]N[/MATH] into odd and even numbers and find two cartesian products:
[MATH][0,1) \times \{2k; k \in N\}[/MATH] and
[MATH][0,1) \times \{2k+1 ;k \in N\}[/MATH] but I am not sure if that is helpful at all.

 

[pka]

Can I have one more question, please?
I have an example here: [MATH]|[0,1)| = |[0,1) \times N| [/MATH]Can you please help me how can I find bijection between these two sets to show that they have equal cardinalities?

Define \(\displaystyle P_0=[0,\tfrac{1}{2})\) now tor \(\displaystyle n>0\) define \(\displaystyle P_n=[\tfrac{n}{n+1},\tfrac{n+1}{n+2})\)
Do you see that \(\displaystyle \bigcup\limits_{k = 0}^\infty {{P_k}} = \left[ {0,1} \right)\). So \(\displaystyle (\forall t\in [0,1), (\exists j)[t\in P_j]\)?
Define \(\displaystyle \Theta: [0,1)\to [0,1)\times N\) by \(\displaystyle \Theta: t\mapsto (t,j)\) .
Can you prove that \(\displaystyle \Theta\) is an injection?
Is there an injection from \(\displaystyle [0,1)\times N\to [0,1)~?\)