  The disjunctive system is a system involving a disjunctive set which is the
union of finitely many polyhedral convex sets. In this paper, we introduce a
notion of the relaxed constant positive linear dependence constraint
qualification (RCPLD) for the disjunctive system. For a disjunctive system, our
notion is weaker than the one we introduced for a more general system recently
(J. Glob. Optim. 2020) and is still a constraint qualification. To obtain the
local error bound for the disjunctive system, we introduce the piecewise RCPLD
under which the error bound property holds if all inequality constraint
functions are subdifferentially regular and the rest of the constraint
functions are smooth. We then specialize our results to the ortho-disjunctive
program, which includes the mathematical program with equilibrium constraints
(MPEC), the mathematical program with vanishing constraints (MPVC) and the
mathematical program with switching constraints (MPSC) as special cases. For
MPEC, we recover MPEC-RCPLD, an MPEC variant of RCPLD and propose the MPEC
piecewise RCPLD to obtain the {error bound property}. For MPVC, we introduce
new constraint qualifications MPVC-RCPLD and the piecewise RCPLD, which also
implies the local error bound. For MPSC, we show that both RCPLD and the
piecewise RCPLD coincide and hence it leads to the local error bound.
