What is the cardinality of the following set: $\{f: \mathbb{R} \rightarrow \{a+b\sqrt[3]4\ | a,b \in \mathbb{Q}\}\}$?
Let $S = \{f: \mathbb{R} \rightarrow \{a+b\sqrt[3]4\ | a,b \in \mathbb{Q}\}\}$.
I already show that $\{f:\mathbb{R}\rightarrow\{0,1\}\} \subset S$, then $|S| \ge |\{f:\mathbb{R}\rightarrow\{0,1\}\}| = 2^c$, and I need to show the rest of the part i.e.: $|S| \le 2^c$, then to conclude that the cardinality of the set $S$ is $2^c$. (please correct me if this part is wrong)
But I get stuck, I don't know how to show that $|S| \le 2^c$. Can you please help me out? Thanks in advance!
You're looking at the set of functins from $\mathbb{R}$ which has size $2^{\aleph_0}= \mathfrak{c}$ to a countable set (as $|\mathbb{Q}^2| = \aleph_0$) which by definition has size $(\aleph_0)^{2^{\aleph_0}} = 2^{\mathfrak{c}}$ by standard cardinal arithmetic : $$ 2^\mathfrak{c} \le \aleph_0^{\mathfrak{c}} \le (2^\mathfrak{c})^\mathfrak{c} = 2^\mathfrak{c}$$