Which has bigger cardinality, $\mathbb Q^\mathbb R$ or $\mathbb R^\mathbb Q$?

Question

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.

Answer 1

$\def\Q{\mathbf Q}\def\R{\mathbf R}$

$\R^{\Q}$ 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 $$ |\Q^\R| \ge |\mathfrak P(\R)| > |\R| = |\R^{\Q}| $$