Linked Questions

1
vote
2answers
1k views

How many infinite subsets of N are there anyway? [duplicate]

I was reading 2 proofs one that the powerset of $ N$ has a higher cardinality than $N$ two a proof that the cardinality of the set of all finite subsets of $N$ has the same cardinality than $N$ ...
0
votes
1answer
165 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 ...
-3
votes
1answer
84 views

prove that $|{\{X\subseteq\mathbb{N}|\ |\mathrm{X|={\aleph_{0}}}\}}| = 2^{\aleph_{0}}$ [duplicate]

I need to prove that: $$|{\{X\subseteq\mathbb{N}|\ |\mathrm{X|={\aleph_{0}}}\}}| = 2^{\aleph_{0}}$$. it's allowed to use the fact that $|P(\mathbb{N})|=2^{\aleph_{0}}=|\mathbb{R}|$ - this is the ...
2
votes
2answers
2k 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 ...
3
votes
3answers
153 views

Cardinal of the set of all subsets of $\mathbb N$ such that both $A$ and the complement are infinite

Given $M=\{ A \in P( \mathbb{N} ) \mid A\text{ and } \overline{\rm A}\text{ are infinite} \}$ What is the cardinal of set $M$? I thought to find the cardinal of $\{ A \in P( \mathbb{N} ) \mid A\...
0
votes
3answers
316 views

Why doesn't this prove that Cantor's Diagonal argument doesn't work?

Let us assume that construction of a number, $z$, which cannot be indexed (or counted) in $\mathbb{N}$, proves by contradiction that the cardinality of the set containing $z$ is greater than $|\mathbb{...
-2
votes
2answers
73 views

Number of one-to-one Function from $\mathbb{N}$ to $\mathbb{N}$ [closed]

How many one-to-one functions are there from $\mathbb{N}$ to $\mathbb{N}$, Where $\mathbb{N}$ is the set of Natural numbers?