Cardinal arithmetic confusion: What is $|\Bbb R|+|\Bbb N|$?

Question

I do not understand how to calculate addition two cardinals. I know that the formula as follows:

if $\alpha$ and $\beta$ are two cardinals, then

$\alpha + \beta= |\{(a,0):a\in \alpha\}\cup\{(b,1):b\in\beta\}|$.

If $\alpha=\mathbb{R}$ and $\beta=\mathbb{N}$, what's cardinals of $\alpha+\beta$?

Answer 1

What is the cardinality of $\Bbb R\times\{0\}\cup\Bbb N\times\{1\}$? You can notice that this is a subset of $\Bbb R\times\{0,1\}$, whose cardinality is $|\Bbb R|+|\Bbb R|$.

This gives you an upper bound, and if you can calculate it, it will also give you a lower bound for the cardinality of $|\Bbb R|+|\Bbb N|$.

As a final remark, you might want to distinguish between a set and its cardinal.

Answer 2

Hint: Try using Schroder-Bernstein to show that $|\mathbb{R}|+|\mathbb{N}|=|\mathbb{R}|$. To do this, let $X=\{(a,0):a\in\mathbb{R}\}\cup\{(b,1):b\in\mathbb{N}\}$; you want to find injections $\mathbb{R}\to X$ and $X\to\mathbb{R}$. The first one should be fairly easy; the second one will require some cleverness.