Abstract: By using the method that a closed fuzzy matroid could be uniquely determined from the basic sequence and the derived matroid equence, the author proposed and proved that a subset family, a sequence, and a surjection from the subset family to the sequence could uniquely determine a closed fuzzy matroid under the conditions of normality,  inheritance, strong monotonic subtraction and growth, and vice versa. The equivalent description established the relationship between the family of ordinary sets and the family of fuzzy independent sets.

Key words: matroid, fuzzy matroid, closed fuzzy matroid, family of induced independent set, mapping of induced independent set, axiom of induced independent set