Linked Questions

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votes
4answers
2k views

show that the set of all 2-element subsets of $\mathbb{N}$ is countable [duplicate]

show that the set of all 2-element subsets of $\mathbb{N}$ is countable could someone guide me through this problem?
0
votes
1answer
3k views

What is the cardinality of a set of all finite subsets of $\Bbb{N}$? [duplicate]

I'm looking for cardinality of $P_{fin}(\Bbb{N})=\{x|x\subset\Bbb{N}$ and $x$ finite$\}$. I was told in my classes that it's $\aleph_0$, but how to prove it?
1
vote
2answers
1k views

Set of all finite subsets of $\mathbb{N}$ is a countable set [duplicate]

Possible Duplicate: Show that the set of all finite subsets of $\mathbb{N}$ is countable. How can I prove in a proper way that the "set of all finite subsets of $\mathbb{N}$ (the set of natural ...
0
votes
3answers
859 views

Cardinality of the set of finite sets in the power set of natural numbers [duplicate]

It is known that $|2^\Bbb{N}|=|\Bbb{R}|$ and that $2^\Bbb{N}$ contains all the subsets of $\Bbb{N}$, just an idea of a question I had and that I would like suggestions on how to tackle. My question ...
2
votes
2answers
1k views

Set of all finite subsets of $\mathbb{Z}_+$ Countable? [duplicate]

Let $A$ be the set of all finite subsets of $\mathbb{Z}_+$, then is $A$ countable? Defiene $I_j$ as the set consisting of all subsets of $\mathbb{Z}_+$ having $j$ elements, then i think $A=\bigcup ...
0
votes
3answers
854 views

How do I proof that the set of finite subsets of $\mathbb{N}$ is countable? [duplicate]

I am working myself into analysis 1 and i came across countable and uncountable sets. Problem 1: I am very confused about the two terms and I would be very thankful if somebody could explain the ...
1
vote
1answer
1k views

how to prove finite subset of a countable set is countable [duplicate]

Let A be a countable set. How to prove that the set $\mathcal A = \{X \subseteq A : |X| = n $ for some $n \in \omega \}$ of all finite subsets of A is countable?. I checked similar questions subset ...
2
votes
3answers
87 views

Is there a bijection $\big\{x:\mathbb{N}\to\{0,1\}:\{i\in\mathbb{N}:x(i)=1\}\text{ is finite}\big\}\to\big\{x:\mathbb{N}\to\{0,1\}\big\}$? [duplicate]

Can we use Cantor's theorem to figure this out? Intuitively the domain seems to be a subset of the codomain, but where do powersets come in? Thanks in advance!
1
vote
1answer
268 views

Prove that co finite set over N is countable [duplicate]

I have $$ L = \{A \in \mathcal P(\mathbb N) | A \text{ co finite over } \mathbb N\}$$ How can I prove that $L$ is countable? Thanks
0
votes
1answer
247 views

Proving the set of finite subsets of $\mathbb{N}$ is countably infinite [duplicate]

So I was given a question that begins like this. Let $P_{\text{fin}}(\mathbb{N})$ be the following set (called the finite power set of $\mathbb{N}$): $$ P_{\text{fin}}(\mathbb{N}) = \{X \...
0
votes
0answers
170 views

Prove: If S is countably infinite set, all finite subsets of S are also countably infinite [duplicate]

Question: Prove that if S is a countably infinite set then all the finite subsets of S are also countably infinite. Where would I start?
0
votes
1answer
128 views

The set of all finite subsets of $\mathbb{N}$ is countable [duplicate]

Can someone explain this in the most straightforward manner possible? I've looked at one other thread on here, but the mathematics was unfortunately too confusing for me. Is there some way I can ...
0
votes
2answers
83 views

Why does this proof that the set of all finite subsets of N is a countable set not work for the set of all subsets of N? [duplicate]

I found this proof in a StackExchange thread and found it pretty understandable and simple: "The other answers give some sort of formula, like you were trying to do. But, the simplest way to see ...
0
votes
1answer
82 views

Constructing a bijection between $\mathbb N $ and the set of all subsets of finite sets of $\mathbb N $ [duplicate]

The answer to this is given, but vaguely and the finishing details are left to be done. I unfortunately am at a loss of an idea as to how to finish it, so was wondering how one would do this last step,...
-1
votes
1answer
124 views

Is there an example of a function $f: \mathbb{Z} \to \{\text{finite subsets of }\mathbb{Z}\}$? [duplicate]

In my last question, I asked for a proof of "Are the set of all finite subsets in $\mathbb{Z}$ countable?" . I had a good answer that showed me that it is an $f: \mathbb{N} \to \{\text{finite subsets ...

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