my question reads as:
Let $\mathcal R[0,1]$ denote the set of all real-valued functions from $[0,1]$ to $\mathbb R$ and let $\mathcal C[0,1]$ denote the set of continuous functions on $[0,1]$.
i) Prove that $|\mathcal C[0,1]|=|\mathbb R$|.
ii) Prove that $|\mathcal R[0,1]|>|\mathbb R$|.
I've been fairly confident in my work with cardinality, but with a set containing functions, I am finding it hard to get a lead.
Thank you in advanced for your help.
Best Answer
I would give you some hints here.
For the first one, you can use the fact that if $f, g\in C [0,1]$ satisfies $f(x) = g(x)$ for all $x\in \mathscr D$, where $\mathscr D$ is a dense subset of $[0,1]$, then $f\equiv g$. Here you can choose $\mathscr D$ to be countable. Then think of $|\{ f:\mathscr D \to \mathbb R\}|$. (The set of all real-valued function from $\mathscr D$).
For the second one, try to construct an injection $P([0,1]) \to R[0,1]$ where $P([0,1])$ is the power set of $[0,1]$. Then use (or prove) $|[0,1]|=|\mathbb R|$.