In this paper, an extensional generalized intuitionistic fuzzy set, called Liu's generalized intuitionistic fuzzy set, is proposed. It is showed that not only Atanassov's intuitionistic fuzzy set but also Mondal & Samanta's generalized intuitionistic fuzzy set is a special case of this new one. Responding the generalized intuitionistic fuzzy set, the generalized intuitionistic fuzzy value (GIFV) is also proposed. Some comparable examples are given, and some important notions and basic algebraic properties of this new generalized intuitionistic fuzzy set and its generalized intuitionistic fuzzy value are discussed.