On the characteristic function of a collection of sets

TL;DR

This paper introduces a simplified formula for the characteristic function of a collection of sets, reducing complexity from exponential to linear in the number of sets, aiding combinatorial analysis.

Contribution

The paper presents a new, simpler expression for the characteristic function of multiple sets, significantly reducing computational complexity.

Findings

Simplified expression requires only n terms instead of 2^n - 1

Facilitates recognition of inclusion-exclusion patterns

Major simplification of normal forms involving characteristic functions

Abstract

The union of a collection of $n$ sets is generally expressed in terms of a characteristic (indicator) function that contains $2^{n}-1$ terms. In this article, a much simpler expression is found that requires the evaluation of $n$ terms only. This leads to a major simplification of any normal form involving characteristic functions of sets. The formula can be useful in recognizing inclusion-exclusion patterns of combinatorial problems.