## Countable sets - question that i dont succeed...

Sequences, counting (including probability), logic and truth tables, algorithms, number theory, set theory, etc.
nir9696
Posts: 1
Joined: Sun Mar 30, 2014 4:21 pm
Contact:

### Countable sets - question that i dont succeed...

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.

Thanks!!!

stapel_eliz
Posts: 1628
Joined: Mon Dec 08, 2008 4:22 pm
Contact:
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?
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"?
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.