Inclusion-Exclusion-Like identities

An expression $E(X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_n)$ built using union, intersection, and complements is called inclusion-exclusion-like if, like the union in the exclusion-inclusion principle, there are constants $c_1,c_2,\mathrm{.}\mathrm{.}\mathrm{.},c_n$ so that for any sequence of sets$A=(A_1,\mathrm{.}\mathrm{.}\mathrm{.},A_n)$ the cardinality of ${ E(A_{1},…,A_{n})}$canbeexpressedas$\left|E(A_{1},…,A_{n})\right|=$${\sum }_{k=1}^n{c}_k{i}_{n,k}(A)$ where $i_{n,k}(A)=\sum_{I\subseteq\overline{n},\left|I=k\right|}^{n}\left|\cap_{i\in I}A_{i}\right|$isthesumofthecardinalitiesofallintersectionsof$k$setsinthesequence$A$\mathrm{.}Inthispaper,weconstruct,fromtheexpression$E$, a set of nonempty subsets of $\{1,\dots ,n\}$ 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 $c_1,c_2,\mathrm{.}\mathrm{.}\mathrm{.},c_n$ 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.