Linked Questions

5
votes
1answer
10k views

Does $\mathbb R^2$ contain more numbers than $\mathbb R^1$? [duplicate]

Possible Duplicate: bijective map from $\mathbb{R}^3\rightarrow \mathbb{R}$ Do the real numbers and the complex numbers have the same cardinality? Does $\mathbb R^2$ contain more numbers than $\...
2
votes
2answers
287 views

How did Cantor demonstrate a bijection from $I=[0,1]$ to $I^n$? [duplicate]

"I See It, but I Don't Believe It." Georg Cantor showed that sets of different dimensions can have the same cardinality; in particular, he demonstrated that there is a bijection between the ...
4
votes
2answers
364 views

Why are the cardinality of $\mathbb{R^n}$ and $\mathbb{R}$ the same? [duplicate]

As I study the first part of abstract algebra, I have a question: why $\left\vert{\mathbb R}\right\vert = \left\vert{\mathbb R^2}\right\vert$? And moreover, I knew the fact that $\left\vert{\mathbb R}...
0
votes
1answer
2k views

Bijection between $[0,1]$ and $[0,1]\times[0,1]$ [duplicate]

I know that $|\mathbb R|=|\mathbb R\times\mathbb R|$, and that $|[0,1]|=|\mathbb R|$, which suggests that $|[0,1]|=|[0,1] \times [0,1]|$ but I would like to know a bijection between the interval and ...
2
votes
1answer
1k views

Is $\mathbb R^2$ equipotent to $\mathbb R$? [duplicate]

I know that $\mathbb N^2$ is equipotent to $\mathbb N$ (By drawing zig-zag path to join all the points on xy-plane). Is this method available to prove $\mathbb R^2 $ equipotent to $\mathbb R$?
3
votes
2answers
154 views

Bijection from $[0,1]^3$ to $[0,1]$? [duplicate]

Is there any bijection from $[0,1]^3$ to $[0,1]$? How can I construct it?
1
vote
3answers
279 views

How to Prove $\mathbb R\times \mathbb R \sim \mathbb R$? [duplicate]

How to prove $\mathbb R\times \mathbb R \sim \mathbb R$? I know you have to split the problem up into two claims, for each direction to prove that it is a bijection, but i don't know how to go much ...
1
vote
2answers
408 views

Bijection from $\mathbb{R} \to \mathbb{R} \times \mathbb{R}$? [duplicate]

Possible Duplicate: bijective map from $\mathbb{R}^3\rightarrow \mathbb{R}$ I know it's possible to produce a bijection from $\mathbb{Z}$ to $\mathbb{Z}\times\mathbb{Z}$, but is it possible to do ...
1
vote
1answer
644 views

Elegant way to make a bijection from the set of the complex numbers to the set of the real numbers [duplicate]

Make a bijection that shows $|\mathbb C| = |\mathbb R| $ First I thought of dividing the complex numbers in the real parts and the complex parts and then define a formula that maps those parts to the ...
2
votes
0answers
820 views

Bijection between $\mathbb{R}\times\mathbb{R}$ and $\mathbb{R}$ [duplicate]

It must be posted somewhere, but I can't find it. I've been working on it for a while too without getting anywhere. Does there exist a bijection between $\mathbb{R}\times\mathbb{R}$ and $\mathbb{R}$? ...
0
votes
1answer
489 views

Bijection between $\mathbb{R}$ and $\mathbb{R}^2$ [duplicate]

I have been thinking for a while whether its possible to have bijection between $\mathbb{R}$ and $\mathbb{R}^2$, but I cant think of a solution. So my question is: is there a bijection between $\...
2
votes
1answer
315 views

Explicit Bijection between Reals and $2 \times 2$ Matrices over the Reals [duplicate]

I understand that the set of all 2x2 matrices over the reals is of the same cardinality as the set of the reals themselves. However, I am curious if a specific, explicit bijection is known to exist, ...
1
vote
1answer
304 views

Bijection between $\mathbb{R}^n$ and $\mathbb{R}^m$ [duplicate]

I've heard that it is possible to present a bijection $\phi : \mathbb{R}^n \to \mathbb{R}^m$ due to Cantor that although not continuous show that those sets have the same cardinality independent of ...
1
vote
1answer
101 views

Is there any bijection between $\mathbb{R}$ and $\mathbb{R}^2$? [duplicate]

Is there any bijection between $\mathbb{R}$ and $\mathbb{R}^2$ ? If have then what is the mapping ? Please define the mapping. They have same cardinality then it is possible to have a bijection ...
0
votes
0answers
121 views

Is it possible to create a bijection between all pairs of reals and a real? [duplicate]

The title basically says it all. Is it possible to associate with each pair of reals, another unique real? I guess you could say I'm looking for functions of two real arguments that return a ...

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