Sets with a negative number of elements (Q1190227)

The author shows how to extend the established combinatorial facts by using ``hybrid'' sets. A hybrid set is defined as follows: Given a universe \(U\), any function \(f: U\to Z\) is called a hybrid set. The value of \(f(x)\) is said to be the multiplicity of the element \(x\). If \(f(x)\neq 0\), we say \(x\) is a member of \(f\) and write \(x\in f\); otherwise, we write \(x\not\in f\). Define the number of elements \(\# f\) to be the sum \(\sum_{x\in U} f(x)\). \(f\) is said to be an \(\# f\) (element) hybrid set.