Proof that the set is countable - my idea The set: S := <mo fence="false" stretchy="false

Desirae Washington 2022-07-15 Answered
Proof that the set is countable - my idea
The set: S := { T   N : T is finite or N T is finite}
Since all finite subsets of N are countable, can i just prove that since N T would simply be { t ; t N t T }... All though i do have the question how could N T be a finite set?
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Answers (2)

Darrell Valencia
Answered 2022-07-16 Author has 10 answers
Explanation:
Let X n be the set of all subsets of N of cardinality n. Then | X n | | N n | = | N | .. Let F be the set of all finite subsets of N .. Then F = n 0 X n .. Can you now finish?
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Jovany Clayton
Answered 2022-07-17 Author has 4 answers
Step 1
Let F = { T N : T  is finite }
I = { T N : N T  is finite }
= { N T : T F }
Thus there is a bijection ϕ : I F where one has
ϕ ( A ) = N A
Step 2
Now suppose we are given a bijection ρ : F N
Consider the map β : S N defined via
β ( s ) = 2 ρ ( s )  if  s F
= 2 ρ ( ϕ ( s ) ) 1  if  s I note that this is a bijection.
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