TITLE: Which has bigger cardinality, $\mathbb Q^\mathbb R$ or $\mathbb R^\mathbb Q$?
QUESTION [5 upvotes]: Which has bigger cardinality, $\mathbb Q^\mathbb R$ or $\mathbb R^\mathbb Q$ ? I think I should use the Schroder-Bernstein theorem but I can't find the necessary injections/ prove that there aren't any.

REPLY [8 votes]: $\def\Q{\mathbf Q}\def\R{\mathbf R}$
$\R^{\Q}$ has cardinality
$$
(2^{\aleph_0})^{\aleph_0} = 2^{\aleph_0 \cdot \aleph_0} = 2^{\aleph_0}
$$
so $|\R^\Q| = |\R|$, but $\Q^{\R}$ has cardinality
$$
|\Q^\R| = \aleph_0^{2^{\aleph_0}} \ge 2^{2^{\aleph_0}},
$$
so
$$
|\Q^\R| \ge |\mathfrak P(\R)| > |\R| = |\R^{\Q}|
$$