Functional dependencies with null values, fuzzy values, and crisp values

Based on the concepts of the semantic proximity, we present a definition of the fuzzy functional dependency, We show that the inference rules for fuzzy functional dependencies, which are the same as Armstrong's axioms for the crisp case, are correct and complete. We also show that dependent constraints with dull values constitute a lattice. Functional dependencies in classical relational databases and null functional dependencies can be viewed as a special case of fuzzy functional dependencies. By applying the unified functional dependencies to the relational database design, we can represent the data with fuzzy values, null values and crisp values under relational database management systems, By using fuzzy functional dependencies, we can compress the range of a fuzzy value and make this fuzzy value "clearer".