In this paper we propose a method to construct more general fuzzy sets using ordinary fuzzy sets as building blocks. We introduce the concept of multi-fuzzy sets in terms of ordered sequences of membership functions. The family of operations T, S, M of multi-fuzzy sets are introduced by coordinate wise t-norms, s-norms and aggregation operations. We define the notion of coordinate wise conjugation of multifuzzy sets, a method for obtaining Atanassov’s intuitionistic fuzzy operations from multi-fuzzy sets. We show that various binary operations in Atanassov’s intuitionistic fuzzy sets are equivalent to some operations in multi-fuzzy sets like M operations, 2-conjugates of the T and S operations. It is concluded that multi-fuzzy set theory is an extension of Zadeh’s fuzzy set theory, Atanassov’s intuitionsitic fuzzy set theory and L-fuzzy set theory.