  We prove that the number of directions contained in a set of the form $A
\times B \subset AG(2,p)$, where $p$ is prime, is at least $|A||B| -
\min\{|A|,|B|\} + 2$. Here $A$ and $B$ are subsets of $GF(p)$ each with at
least two elements and $|A||B| <p$. This bound is tight for an infinite class
of examples. Our main tool is the use of the R\'edei polynomial with
Sz\H{o}nyi's extension. As an application of our main result, we obtain an
upper bound on the clique number of a Paley graph, matching the current best
bound obtained recently by Hanson and Petridis.
