I'm trying to show that $\{(x,y)\in \mathbb{R}^2: x^2+y^2<1\}$ and $\mathbb{R}^2$ have the same cardinality.
It has been given to me that for any finite interval (open or closed), it has the same cardinality as the reals. Hence, it would be nice for me to use something like: $|A \times B|=|C \times D|$ if $|A| = |C|$ and $|B|=|D|$.
I highly doubt that this holds true for uncountable cardinalities. My approach to this question is to show $|\mathbb{R}\times\mathbb{R}|=|[0, 1) \times [0, 2 \pi )|$ which will then allow me to use the $r$ and $\theta$ way to create a one-to-one relation to the open unit disk.
Any help is appreciated, and so is any criticism. Thank you very much.