Three theorems of interval fuzzy set

The purpose of this paper lies in the combination of fuzzy set and interval grey set, namely, considering the fuzzy set theory, rough set theory and grey system theory. The paper then presents a unified form (or symbol system) which can be more useful to describe a wide variety of theories.

Considering the lower bounds and upper bounds of interval grey set are classical set, this paper draws lessons from the classical set to fuzzy set transition mode which was advanced by Professor Zadeh. The paper puts forward the concept of interval fuzzy set, and then relaxes the constraints of the lower bound of interval fuzzy set to make it more widely used in actual uncertain problems.

For the examples which have been given in the paper, interval fuzzy set and its three theorems are proved practical and flexible.

The theorems proved in the paper can be used as a useful extension to the classical fuzzy set, which can also be applied to fuzzy clustering, fuzzy classification, fuzzy pattern recognition and so on.

The paper puts forward the concept of interval fuzzy set and the three theorems of interval fuzzy set and proves their practicability and flexibility. The theorems can be used for further study on interval fuzzy set.