Decomposing the domain of a function into parts has many uses in mathematics.
A domain may naturally be a union of pieces, a function may be defined by
cases, or different boundary conditions may hold on different regions. For any
particular problem the domain can be given explicitly, but when dealing with a
family of problems given in terms of symbolic parameters, matters become more
difficult. This article shows how hybrid sets, that is multisets allowing
negative multiplicity, may be used to express symbolic domain decompositions in
an efficient, elegant and uniform way, simplifying both computation and
reasoning. We apply this theory to the arithmetic of piecewise functions and
symbolic matrices and show how certain operations may be reduced from
exponential to linear complexity.
