Abstract: The real number weight of the element of a countable set was defined. By using elementary and combinatorial transformation methods, the calculating formulae of the real number weight of a series of sets that consist of some elements which meet under the limited groups of conditions on countable sets were studied. Some new theorems of the generalized principle of inclusionexclusion were given. The results can improve the regular principle of inclusionexclusion, which shows that the theorems are more elementary and simpler than other existing theorems. Finally, two typical cases to explain the application of the generalized principle of inclusionexclusion were given.

Key words: countable set, real number weight, set characteristic function, generalized inclusionexclusion principle