Countability problem If we have a line in a plane. Does the line have $\#\Bbb R$ points? And if yes, How many points does the plane have? More than $\#\Bbb R$ or equal to $\#\Bbb R$and why? I am talking about $\Bbb R^3$ here.
 A: It has size equal to that of $\mathbb{R}$; the bijection is given as follows:
If the line is not vertical, then projection onto the $x$-axis gives a bijection.  Otherwise, it is vertical and you can project onto the $y$-axis, or even just translation works.
EDIT: I will make use of a theorem called the Schroeder-Cantor-Bernstein Theorem, which states the following:

Let $A$ and $B$ be two non-empty sets.  If there exist injections $f:A\rightarrow B$, $g:B\rightarrow A$, then there exists a bijection $h:A\rightarrow B$.

In effect, it says that if $|A|\leq |B|$ and $|B|\leq |A|$, then $|A|=|B|$.
Now, the injection from $\mathbb{R}$ to $\mathbb{R}^2$ is quite simple: take every real number $a$ to the ordered pair $(a,0)$.
The injection from $\mathbb{R}^2$ to $\mathbb{R}$ requires slightly more.  We first recognize that $\mathbb{R}$ can be put into bijection with the set of all infinite decimals that do not end in an infinite string of 0's with integer part zero (i.e. the elements of $(0,1)$; let me know if you'd like an explicit bijection to show that $|\mathbb{R}|=|(0,1)|$).  (Of course, any base could be picked.)  So given an ordered pair $(a,b)$, we write it as $(0.a_1a_2a_3\ldots a_n\ldots,0.b_1b_2b_3\ldots b_n\ldots)$, and define our injection by sending it to $0.a_1b_1a_2b_2a_3b_3\ldots a_nb_n\ldots$.  
Thus, by applying Schroeder-Cantor-Bernstein, we see that $\mathbb{R}^2$ has the same cardinality as $\mathbb{R}$.  Let us say that the bijection is $f:\mathbb{R}^2\rightarrow \mathbb{R}$.
The above proof could be used to show that any $\mathbb{R}^n$ has the same cardinality as $\mathbb{R}$ inductively; for example, for $\mathbb{R}^3$ we could use $f\times id_\mathbb{R}$ to map $\mathbb{R}^3$ to $\mathbb{R}^2$ in a one-to-one onto fashion, and then compose this with $f$ to give us the bijection with $\mathbb{R}$.  
