Find the cardinality of the set of all points in the plane which have one rational and one irrational coordinate. Justify you answer.
My thoughts so far. We know that $\mathbb Q$, the set of all rational numbers, has a cardinality of $\aleph_0$. Also, since the set of irrational numbers is just the reals with the rationals removed, thus the set of irrational numbers has a cardinality of $\mathfrak c$ (continuum). Therefore, it seems logical to conclude that the ordered pair also has a cardinality of $\mathfrak c$.
I'm fairly sure this would justify the answer. However, is there a way to prove this using either a bijection or the Cantor-Bernstein theorem? Thanks in advance.