Linked Questions

7
votes
6answers
7k views

Is the set of all pairs of natural numbers countable? [duplicate]

Say that $\Bbb N \times \Bbb N$ is the set of all pairs $(n_1, n_2)$ of natural numbers. Is it countable? My hypothesis is yes it is countable because sets are countable. But I am unable to come up ...
0
votes
2answers
3k views

Union, intersection and Cartesian product of countable sets is countable [duplicate]

Possible Duplicate: Countable Sets and the Cartesian Product of them Sum of two countably infinite sets I want to solve a problem, this problem is the following: Prove that if the sets $A$ and ...
4
votes
2answers
1k views

Show that $\mathbb{Q}\times \mathbb{Q}$ is denumerable [duplicate]

I am new to functions and relations, and with some concepts I am not so familiar. I have a question in an homework: Show that $\mathbb{Q} \times\mathbb{Q}$ is denumerable. From what I understood, ...
0
votes
4answers
2k views

How do I show that the set of natural number N has the same size as NxN? [duplicate]

How do I show that the set of natural numbers $\mathbb{N}$ has the same size as $\mathbb{N}\times\mathbb{N}$? I know that for two sets to have the same size, there must be an injection from the set $\...
2
votes
1answer
2k views

Is $\mathbb Q \times \mathbb Q $ a denumerable set? [duplicate]

How can one show that there is a bijection from $\mathbb N$ to $\mathbb Q \times \mathbb Q $?
0
votes
3answers
318 views

Bijection for a function $ \mathbb{Z}^+ \times \mathbb{Z}^+$ to $\mathbb{Z}^+ $ [duplicate]

Possible Duplicate: Countable Sets and the Cartesian Product of them Consider the following question: Describe a function $ \mathbb{Z}^+ \times \mathbb{Z}^+$ to $\mathbb{Z}^+ $ that is one-to-...
2
votes
2answers
347 views

Cardinality of $\omega^2$ [duplicate]

I know $ \omega ^ 2 $ is countable, but I'm unable to find a bijection from $ \omega * \omega \rightarrow \omega $ This should be simple, but I'm very stuck.
1
vote
2answers
470 views

If $A$ and $B$ are countable sets, show that $A \times B$ is countable [duplicate]

My question is If $A$ and $B$ are countable sets, show that $A \times B$ is countable. I know the definitions to be a countable set are: A set $A$ is countable if $A$ is finite or countably infinite....
-2
votes
2answers
210 views

Is the Cartesian Product of a finite number of infinitely countable sets countable? [duplicate]

Let $S=A_1 \times A_2 \times ...\times A_n$ Is S countable? And how do I prove it? I think the answer is yes because $A_1 \times A_2$ creates an infinite table so for n sets we would have an ...
-2
votes
2answers
92 views

$\Bbb N$x$\Bbb N$ is countably infinite [duplicate]

How can I prove $\Bbb N$x$\Bbb N$ is countably infinite? Is the proof in some book about sets? Somebody help please.
-1
votes
1answer
65 views

Countability of Different Sets [duplicate]

(a) Prove that $N \times N$ is a countable set (b) Let T be the set of two element subsets of N. Prove that T is countable. This is a question in my exam review package. I missed the lesson on ...
0
votes
1answer
68 views

I need help with proofs pertaining to countability [duplicate]

Possible Duplicate: Countable Sets and the Cartesian Product of them Inductive Proof of a countable set Cartesian product Let $A$ and $B$ be countable sets. (a) Show that $A \times B$ is ...
39
votes
7answers
61k views

Prove that the union of countably many countable sets is countable.

I am doing some homework exercises and stumbled upon this question. I don't know where to start. Prove that the union of countably many countable sets is countable. Just reading it confuses me. ...
5
votes
5answers
7k views

Proving $\mathbb{N}^k$ is countable

Prove that $\mathbb{N}^k$ is countable for every $k \in \mathbb{N}$. I am told that we can go about this inductively. Let $P(n)$ be the statement: “$\mathbb{N}^n$ is countable” $\forall n \in \...
5
votes
2answers
4k views

Is the set of polynomial with coefficients on $\mathbb{Q}$ enumerable?

Using the definition of enumerability of sets: A non-empty set B is enumerable iff there is a bijection $f:\mathbb{N}\supset L \rightarrow B$. So, I have to prove that the set of polynomial of one ...

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