  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.
