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Matroid theory is a useful tool for the combinatorial optimization issue in data mining, machine learning and knowledge discovery. Recently, combining matroid theory with rough sets is becoming interesting. In this paper, rough set approaches are used to investigate an important class of matroids, transversal matroids. We first extend the concept of upper approximation number functions in rough set theory and propose the notion of generalized upper approximation number functions on a set system. By means of the new notion, we give some necessary and sufficient conditions for a subset to be a partial transversal of a set system. Furthermore, we obtain a new description of a transversal matroid by the generalized upper approximation number function. We show that a transversal matroid can be induced by the generalized upper approximation number function which can be decomposed into the sum of some elementary generalized upper approximation number functions. Conversely, we also prove that a generalized upper approximation number function can induce a transversal matroid. Finally, we apply the generalized upper approximation number function to study the relationship among transversal matroids. [ABSTRACT FROM AUTHOR]