Proving the Infinity of C\B: A Contradiction Method

In summary, if C is an infinite set and B is a finite set, then C\B is an infinite set. This can be proven by contradiction, assuming that C\B is finite and using the fact that there exist bijective functions from C\B to Nk and B to Nl for some k and l respectively. By using these two bijective functions, we can define a third bijective function from C\B to N(k-l), which implies that C is finite. This contradiction proves that C\B must be infinite.
  • #1
Punkyc7
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0
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[itex]\alpha[/itex] 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?
 
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  • #2
Hi Punkyc7! :smile:
Punkyc7 said:
I have Assume C\B is a finite set. Then there exist a function[itex]\alpha[/itex] 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. :wink:
 
  • #3
so I would say something like there exist a a function[itex]\beta[/itex] 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[itex]\chi[/itex] that is bijective to N(k-l) which would imply c is finite [itex]\rightarrow\leftarrow[/itex]

Does that work?
 
  • #4
Hi Punkyc7! :smile:

(just got up :zzz: …)

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

1. What is the difference between infinite sets and finite sets?

Infinite sets contain an uncountable number of elements, while finite sets contain a finite number of elements.

2. Can an infinite set be a subset of a finite set?

Yes, an infinite set can be a subset of a finite set. For example, the set of all natural numbers is infinite, but it is a subset of the finite set of all real numbers.

3. How do you know if a set is infinite or finite?

A set is infinite if it has an uncountable number of elements, meaning that it cannot be put into a one-to-one correspondence with the set of natural numbers. A set is finite if it has a finite number of elements, which can be counted.

4. Are there different types of infinite sets?

Yes, there are different types of infinite sets. For example, the set of all real numbers is a larger type of infinity than the set of all natural numbers.

5. Can two infinite sets have the same number of elements?

No, two infinite sets cannot have the same number of elements. If they did, they would be considered the same size and thus not infinite.

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