Show that $\mathbb{R}$ is a disjoint union of $\mathfrak{c}$ sets of cardinal $\mathfrak{c}$ 
Show that $\mathbb{R}$ is a disjoint union of $\mathfrak{c}$ sets of cardinal $\mathfrak{c}$, where $\mathfrak{c} = | \mathbb{R} | = 2^{\aleph_0}$.

I find this problem very interesting and very challenging at the same time. Since any set of sets with cardinality $\mathfrak{c}$ I can think about are not disjoint. There is probably a specific definition of this $\mathfrak{c}$ sets of cardinal $\mathfrak{c}$ which immediately solves the problem. However, I would like to understand how can I possibly think about this. 
 A: HINT: Show that $\Bbb R$ and $\Bbb{R^2}$ have the same cardinality. Now show that $\Bbb R^2$ is the union of $\frak c$ sets of size $\frak c$.
(Note that the proof itself generalizes to any infinite set $A$ such that $A$ and $A^2$ have the same cardinality; which is all of them if you assume the axiom of choice.)
A: We can  construct an uncountable collection of paiwise disjoint sets of real  numbers using decimal expansions.
The collection of subsets $A$ of positive integers such that  both $A$ and complement of $A$ are infinite is an uncountable collection. 
For such a set $A$, denote by $S_A$ the set of all those real numbers whose decimal expansion has digit $0$ exactly  at positions specified by $A$ and nowhere else.
$S_A$ is an uncountable set, because there are infinitely many positions for non-zero numbers where we can place other numbers arbitrarily (for example  numbers using the digits 1 and 2 alone in those positions are uncountable as $A^C$ is an infinite set by choice.)
A: Another decimal-type solution.
Any real number $x$ can be written in "binary" form
$x = n+\sum_{i=1}^\infty a_i 2^{-i}$, where $n \in \mathbb Z$, and $a_i \in \{0,1\}$.  Let us choose non-terminating expansions, if there is more than one expansion.
For $0 \le t \le 1$, let $A_t$ be the set of all reals $x = n+\sum_{i=1} a_i 2^{-i}$ such that
$$
\limsup_{N \to \infty}\frac{1}{N}\sum_{i=1}^N a_i = t
$$
Then each $A_t$ has cardinal $\mathfrak c$ and $\mathbb R = \bigcup_{0 \le t \le 1} A_t$
