The principle of inclusion and exclusion is one of basic tools of classical set
theory. (It has different names; a trivial example is the following: How many
people are there in the world? Sum up the citizens of all states, subtract
those
with double citizenship, add those with triple citizenship, etc.)
Many previous papers have presented its generalizations to fuzzy sets,
intervalvalued fuzzy sets, or IFsets (Atanassov’s intuitionistic fuzzy
sets) with some set operations, where cardinality is replaced by a probability
measure. We clarified which operations satisfy the principle of inclusion and
exclusion.
The material is published in the paper
Principles of inclusion and exclusion for intervalvalued fuzzy sets and
Fuzzy Sets and Systems, Volume 324, Pages 6073
https://www.sciencedirect.com/science/article/pii/S0165011416303402/pdfft?md5=5488ade44b2967ddd27694063e4cbe06&pid=1s2.0S0165011416303402main.pdf
