We consider the Dirichlet problem for a class of fully nonlinear elliptic equations on a bounded domain $\Omega$. We assume that $\Omega$ is symmetric about a hyperplane H and convex in the direction perpendicular to H. Each nonnegative solution of such a problem is symmetric about H and, if strictly positive, it is also decreasing in the direction orthogonal to H on each side of H. The latter is of course not true  if the solution has a nontrivial nodal set. In this paper we prove that for a class of domains, including for example all domains which are convex (in all directions), there can be at most  one nonnegative solution with a nontrivial nodal set. For general domains,  there are at most finitely many such solutions.