maths question - cardinality of the real no.s?

Question

is the cardinality of the set of real numbers equal to the cardinality of the set of real numbers cross itself?
ie is card(R) = card(RxR)?

i can find an injection from R to RxR (which implies card (R) <= card (RxR) ) but not a bijection.

Answer 1

I went a-lookin, cuz I didn't know the answer.

Here's what I found:

Source: http://www2.cs.uh.edu/~robertr/math.html

One to one mapping from RxR to R:
any number in RxR we call (A,B)
A= a1a2a3a4a5a6..... (where aX is a digit)
B= b1b2b3b4b5b6.... (where bX is a digit)

Now to map (A,B) in RxR to (C) in R:
C=a1b1a2b2a3b3a4b4a5b5.... etc...

So there is a unique number in R for each number in RxR,
so they are in 1-1 correspondence,
Therefore, they are the same cardinality.
And since you can also do a similar thing for all points in R to
points between 0 and 1, and for RxRxR.....xR, we see that:

you can map every point in the universe at every time interval to the points between your thumb and forefinger.

---------------------
I know, this is more of an algorithm than a bijection.

There's a significant discussion of this issue on a number of sites:

http://www.google.com/search?hl=en&lr=&safe=off&rls=DELA%2CDELA%3A2005-51%2CDELA%3Aen&q=bijection+cardinality+of+RxR

Answer 2

You can prove that Z and Q have the same cardinality so I would look at the proof since for R and RXR it may be similar.

Answer 3

It's easy to find injections both ways. So there is a theorem I think by Bernstein, which says that there is a bijection between the two. But it doen't tell you how to construct it.