I'm trying to calculate the cardinality of $A$: $$A=\{L \subseteq \text{$Σ^*$}| \text{$L^R$}=L \}$$ Where Σ is a finite alphabet . The hint in my book says to use "Binary Representation of real numbers". I don't see what can I do with that hint, any help will be appreciated.
*$$\text{$L^R$}=\{W^R|W \in L\}$$
If $\Sigma = \emptyset$, then $\Sigma^* = \{1\}$ (where $1$ denotes the empty word) and the two languages of $\Sigma^*$, namely $\emptyset$ and $\{1\}$, are closed under reversal. Thus, if $\Sigma = \emptyset$, the answer is $2$.
If $\Sigma$, is a one-letter alphabet, say $\Sigma = \{a\}$, then again, every language of $\Sigma^*$ is closed under reversal. Thus in this case, the answer is $\text{Card}(\cal{P}(\mathbb{N}))$.
If $1 < |\Sigma| < \infty$, then $\Sigma^*$ is countable, and thus $$ \textrm{Card}(\mathcal{P}(\Sigma^*)) = \textrm{Card}(\mathcal{P}(\mathbb{N})) $$ Now since $A$ contains the set $\mathcal{P}(a^*)$ for each letter $a$ of $\Sigma$, the cardinality of $A$ is equal to $\text{Card}(\cal{P}(\mathbb{N})) = 2^{\aleph_0}$.
Note. I have no idea how to use the hint.