  Inter-relations between random matrix ensembles with different symmetry types
provide inter-relations between generating functions for the gap probabilites
at the spectrum edge. Combining these in the scaled limit with the exact
evaluation of the gap probabilities for certain superimposed ensembles with
orthogonal symmetry allows for the exact evaluation of the gap probabilities at
the hard and soft spectrum edges in the cases of orthogonal and symplectic
symmetry. These exact evaluations are given in terms of Painlev\'e
transcendents, and in terms of Fredholm determinants.
