Inclusion-Exclusion-Like identities

TL;DR

This paper characterizes inclusion-exclusion-like identities involving unions, intersections, and complements, providing conditions and methods to determine their coefficients based on a constructed characteristic set.

Contribution

It introduces a set-based characterization and a criterion for identifying inclusion-exclusion-like identities, along with a method to compute their coefficients.

Findings

Provides a necessary and sufficient condition for inclusion-exclusion-like identities.

Introduces the concept of a characteristic set for such expressions.

Develops a method to determine the constants in the identities.

Abstract

An expression $E(X_{1},...,X_{n})$ built using union, intersection, and complements is called inclusion-exclusion-like if, like the union in the exclusion-inclusion principle, there are constants $c_{1},c_{2},...,c_{n}$ so that for any sequence of sets${ A=(A_{1},...,A_{n})}$ the cardinality of ${ E(A_{1},...,A_{n})}$ can be expressed as $\left|E(A_{1},...,A_{n})\right|=$$\sum_{k=1}^{n}c_{k}i_{n,k}(A)$ where $i_{n,k}(A)=\sum_{I\subseteq\overline{n},\left|I=k\right|}^{n}\left|\cap_{i\in I}A_{i}\right|$ is the sum of the cardinalities of all intersections of $k$ sets in the sequence $A$. In this paper, we construct, from the expression $E$, a set of nonempty subsets of $\left\{ 1,\ldots,n\right\} $ called the characteristic set of $E$, and using this set to give a necessary and sufficient condition for the expression to be inclusion-exclusion-like . Furthermore, we give a method for determining the constants $c_{1},c_{2},...,c_{n}$ in the expression for the cardinality of ${ E(A_{1},...,A_{n})}$ when it exists. The content of the paper is illustrated by a simple detailed example given in the introduction.