Getting results with negative thinking

Abstract

Given a universe of discourse $U$, a {\em multiset} can be thought of as a function $M$ from $U$ to the natural numbers ${\bf N}$. In this paper, we define a {\em hybrid set} to be any function from the universe $U$ to the integers ${\bf Z}$. These sets are called hybrid since they contain elements with either a positive or negative multiplicity. Our goal is to use these hybrid sets {\em as if} they were multisets in order to adequately generalize certain combinatorial facts which are true classically only for nonnegative integers.