cardinality of $\mathbb R$ is the same as the cardinality of $\mathbb R^2$

Question

A problem in my homework is to prove the cardinality of $\mathbb R$ and the cardinality of $\mathbb R^2$ is the same. I'm in my first semester and I have no clue how to do this, do I need the axiom of choice to prove this? Can I find a bijection, or must I use Cantor-Schroeder-Bernstein? Apparently if we assume choice we can go further and prove the cardinality of any set $A$ is is equal to the cardinality of $A^2$ thank you very much, I need help.

Regards.

Here's the answer. http://math.stackexchange.com/questions/243590/bijection-from-mathbb-r-to-mathbb-rn — Math.StackExchange, Oct 30 '14 at 01:07
Do you know already that $|\mathbb R|=|\mathcal P(\mathbb N)|$? — hmakholm left over Monica, Oct 30 '14 at 01:07
no, that question literally came out of the blue, the rest of the homework is full of really simple questions. Thanks — Asinomás, Oct 30 '14 at 01:09
If you're using Bernstein, one direction is trivially easy. For the other direction, a hint: Cantor's proof (or one of them) was by interlacing decimal digits ... — hmakholm left over Monica, Oct 30 '14 at 01:12
Thanks I think I got it. just one question, is this independant of choice? — Asinomás, Oct 30 '14 at 01:15
Yes, it's very independent of choice. The Cantor-Bernstein theorem is independent of choice as well. — Asaf Karagila, Oct 30 '14 at 13:39

score 0 · Accepted Answer · answered Dec 17 '14 at 17:46

I just wanted to get the selfie badge. The solution I found at the end is the following:

We want an injection from $\mathbb R^2 $ to $\mathbb R$ we take a surjection from $\mathbb R$ to $\mathbb R^2$ and what we want is the right inverse.

As an example of a surjection I took the following: let the number in $\mathbb R$ be $(-1)^n(k+.d_1d_2d_3\dots)$ if $k$ is not of the form $2^a3^b5^c7^d$ send it to zero. if it is send it it to $((-1)^ad_1d_3d_5\dots,((-1)^bd_2d_4d_6\dots)$. And leave $c$ and $d$ digits before adding the decimal point.

cardinality of $\mathbb R$ is the same as the cardinality of $\mathbb R^2$

1 Answers1