Cardinality of sets of functions $\mathbb{R}\to \mathbb{N}$ and $\mathbb{N}\to \mathbb{R}$.

Question

Let $B^A$ denote the set of all functions $A \to B$.

Prove that $\left|\mathbb{R}^\mathbb{N}\right|<\left|\mathbb{N}^\mathbb{R}\right|$.

Answer 1

$\ \left|\mathbb{R}^\mathbb{N}\right|<\left | 2^{(\mathbb{R}^\mathbb{N})} \right |\le \left | \mathbb N^{(\mathbb{R}^\mathbb{N})} \right |= \left | \mathbb N^{\mathbb{R}} \right |$.

Answer 2

The set of functions from $A$ to $B$ is denoted $B^A$ because its cardinality is $|B|^{|A|}$ (there are $|A|$ choices to make in specifying such a function, each with $|B|$ options). In particular $$|\Bbb R|^{\Bbb N}=c^{\aleph_0}=2^{\aleph_0^2}=2^{\aleph_0}=c,\,|\Bbb N|^{\Bbb R}=\aleph_0^c=2^c>c.$$Let's explain that last equation:$$2\lt\aleph_0\lt 2^{\aleph_0}\implies 2^c\le\aleph_0^c\le 2^{\aleph_0 c}=2^c.$$