Linked Questions

0
votes
4answers
5k views

Construct a bijection between $\mathbb{Z}^+\times \mathbb{Z}^+$ and $\mathbb{Z}^+$ [duplicate]

Possible Duplicate: Bijecting a countably infinite set $S$ and its cartesian product $S \times S$ The cartesian product $\mathbb{N} \times \mathbb{N}$ is countable I wanna know if $\mathbb{Z^+}\...
1
vote
2answers
897 views

Bijective Function from N to N x N [duplicate]

I have an idea but I don't know how to formalize my idea in a function. That's what n should give me back as (x,y): ...
4
votes
1answer
1k views

Bijection between natural numbers $\mathbb{N}$ and natural plane $\mathbb{N} \times \mathbb{N}$ [duplicate]

I know that is possible to build a bijection between the set of natural numbers $\mathbb{N}$ and the natural plane (the cartesian product of $\mathbb{N}$ by itself, $\mathbb{N} \times \mathbb{N} = \...
1
vote
2answers
421 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....
0
votes
1answer
261 views

Bijection between $\mathbb N^2$ and $\mathbb N$ [duplicate]

A question regarding a bijection between $\mathbb N^2$ and $\mathbb N$. I know about cantor pairing function but I wanted to ask about a bijection I have seen around the site which is $n=2^{u-1}(2v-1)$...
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 ...
4
votes
5answers
2k views

How to generate unique id from each element in matrix?

I'm coming from the programming world , and I need to create unique number for each element in a matrix. Say I have a $4\times4$ matrix $A$. I want to find a simple formula that will give each of the $...
1
vote
5answers
893 views

Existence of a function from $f : \mathbb Z^2 \rightarrow \mathbb Z$

I have a problem with the following question: Does there exist a function $f : \mathbb Z^2 \rightarrow \mathbb Z$ that is one-to-one and onto, and hence invertible? (If yes, then $\mathbb Z^2$ ...
3
votes
2answers
203 views

A bijective mapping from $\mathbb N^k$ to $\mathbb N$?

Having $k$ numbers $N_i\in\mathbb{N}$, I'm looking for a bijective mapping $f:\mathbb{N}\times\ldots\times\mathbb{N}\rightarrow\mathbb{N}$ So that $f^{-1}\left(N_0\right)=\left(N_1,\ldots,N_k\...
1
vote
1answer
582 views

Prove countable set function: natural numbers and pairs of natural numbers

Can someone please explain me the proof, where there is a 1-1 Correspondence between the set of natural numbers and the set of all pairs of natural numbers How can the below data be one-to-one ...
1
vote
1answer
146 views

On the bijection between the naturals and a countable number of countable sets.

I am assuming we are utilizing the axiom of choice because I read in the suggested questions that may have my answer that it's needed for the following bijection to work. Suppose I am given a ...
2
votes
2answers
78 views

One-to-one functions between vectors of integers and integers, with easily computable inverses

I'm trying to find functions that fit certain criteria. I'm not sure if such functions even exist. The function I'm trying to find would take vectors of arbitrary integers for the input and would ...
1
vote
1answer
32 views

Combinatorial problem on finding the index associated to an edge of a complete graph

Ok so here is a combinatorial problem that I thought of. Suppose N is in $\mathbb N$ such that $N>1$, then there is a way to count (set an index) to all pairs $(i,j) \in \{1,\dots,\mathbb N\}\...