Proving the Infinity of C\B: A Contradiction Method

[Punkyc7]

If C is an infinite set and B is a finite set then C\B is an infinite set.

C\B means the complement of B relative to C


Ok so I was thinking of doing this by contradiction.

I have Assume C\B is a finite set. Then there exist a function$\alpha$ that is bijective from C\B to Nk for some k.

Now this is were I am stuck. Obviously if you have something that is infinite and you take away some finite number of thigs is still going to be infinite but how do you write that mathmatically?

 

[tiny-tim]

Hi Punkyc7!

I have Assume C\B is a finite set. Then there exist a function$\alpha$ that is bijective from C\B to Nk for some k.

You haven't yet used the fact that B is finite, and so also has a bijective function.

 

[Punkyc7]

so I would say something like there exist a a function$\beta$ that is bijective to Nl for some l.

Now how does that lead you to say c is finite for the contradiction?
Can you say there exist a function$\chi$ that is bijective to N(k-l) which would imply c is finite $\rightarrow\leftarrow$

Does that work?

 

[tiny-tim]

Hi Punkyc7!

(just got up :zzz: …)

yes, if you can use the first two bijective functions to define a third bijective function, that will prove it. ​