I’m trying to intuitively reason why the cardinality of set of all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ is larger than that of the reals.
I’ve read this explanation:
For any real number there are $|\mathbb{R}|$ possible other reals to get mapped to. And since there are $|\mathbb{R}|$ reals, the total possibilities are $|\mathbb{R}|^{|\mathbb{R}|}$.
I understand why this is larger than $|\mathbb{R}|$, but I don’t understand why this is also the amount of functions. Can someone explain?
