Let $a, b, c, d$ be real numbers such that $a < b$ and $c < d$. Prove that $|[a,b]$ x $[c,d]| = c$. This $c$ on the end is the 'cardinality of the continuum' which means that the set has the same cardinality as the set of real numbers.
I am a bit confused as to where to start with this. Does this mean that the elements of the set are {c,a}, {c,b}, {a,d}, {b,d} where $a, b, c, d$ are real numbers and $a < b$ and $c < d$?
There is a theorem that if $a, b, c, d$ are real numbers and $a < b$ and $c < d$ then $(a,b]$ and $(c,d]$ have the same cardinality. Would this be of any use?