Calculate the cardinality of $\mathcal P_\mathfrak{c}(\mathbb R)=\{\mathcal R \in \mathcal P(\mathbb R) \, /\, \#(\mathcal R)=\mathfrak{c}\}$.
I thought that instead of $\mathcal P_\mathfrak{c}(\mathbb R)$ I should think in $\mathcal P_c([0,1))$ so i can work with the representation of $x \in [0,1)$ in the binary numeral system but I don't know how to do a bijection with that set. Of course, I am thinking that $\mathcal P_c(\mathbb R)$ has the cardinality of $\mathbb R$. (Aclaration: $\mathfrak{c}=\#\mathbb R)$
The cardinality of $\mathcal{P}_\mathfrak{c}(\mathbb{R})$ is $2^\mathfrak{c}$.
Clearly we have $\mathcal{P}_\mathfrak{c}(\mathbb{R}) \subseteq \mathcal{P}(\mathbb{R})$ so that $\left|\mathcal{P}_\mathfrak{c}(\mathbb{R})\right|\le 2^\mathfrak{c}$.
On the other hand, we have $X\cup (0,\infty) \in \mathcal{P}_\mathfrak{c}(\mathbb{R})$, where $X$ is an arbitrary subset of $(-\infty,0)$. But the number of such subsets is $2^\mathfrak{c}$ since $|(-\infty,0)| = \mathfrak{c}$. Hence $\left|\mathcal{P}_\mathfrak{c}(\mathbb{R})\right| \ge 2^\mathfrak{c}$.