Abstract
The combination of the rough sets theory with in-tuitionistic fuzzy sets is a novel theory and a flourish research community in dealing with uncertaintydecision or incomplete and imprecise information. This paper studies the primary theory of intuitionistic fuzzy rough sets over two universes. Firstly, we establish the intuitionistic fuzzy rough sets model over two universes with a constructive approach. Then, we study the properties of lower and upper approximation operators of intuitionistic fuzzy rough sets in the fuzzy approximation space of two universes. At the same time, we introduce the definition of the level cut sets for intuitionistic fuzzy rough sets over two universes. Finally, we establish the decomposition theorem for intuitionistic fuzzy rough sets over two universes according to the definitions of level cut sets
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