Countable sets - question that i dont succeed...

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Countable sets - question that i dont succeed...

Postby nir9696 on Sun Mar 30, 2014 4:56 pm

1. K is the set of all finite subsets in N: A is finite set in N and K = {A in P(N)}. prove that K is a countable set.
2. we say that A is co-finite set in N if A is an Infinite set in N and A' (the complement of A) is a finite set in N. L is the set of all co-finite subsets in N: A is co-finite set in N and L = { A in P(N)}. prove that L is a countable set.
3. M is the set of the all subsets in N which are infinite and their complements (A') are infinite. prove that M is uncountable set. then find the cardinal number of M.



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Postby stapel_eliz on Tue Apr 01, 2014 11:23 am

nir9696 wrote:1. K is the set of all finite subsets in N: A is finite set in N and K = {A in P(N)}. prove that K is a countable set.

The notation "P(N)" standardly means "the power set of N". Is that what you mean here?

nir9696 wrote:2. we say that A is co-finite set in N if A is an Infinite set in N and A' (the complement of A) is a finite set in N. L is the set of all co-finite subsets in N: A is co-finite set in N and L = { A in P(N)}. prove that L is a countable set.

When you say "A is a set in N", do you mean that "A is a subset of N"?

nir9696 wrote:3. M is the set of the all subsets in N which are infinite and their complements (A') are infinite. prove that M is uncountable set. then find the cardinal number of M.

Counter-example (that is, a dis-proof): Let N be a finite set. Then there are no infinite subsets. Then M is the empty set, and is certainly countable.

Please reply showing your work on the other two exercises. Thank you. :wink:
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