Set-valued data are a common type of data for characterizing uncertain and missing information. Traditional dominance-based rough sets can not efficiently deal with large-scale set-valued decision tables and usually neglect the disjunctive semantics of sets. In this paper, we propose a general framework of feature selection and approximate reasoning for large-scale set-valued information tables by integrating quantitative rough sets and dominance-based rough sets. Firstly, we define two new partial orders for set-valued data via the conjunctive and disjunctive semantics of a set. Secondly, based on α-disjunctive dominance relation and α-conjunctive dominance relation defined by the inclusion measure, we present α-dominance-based quantitative rough set models for these two types of set-valued decision tables. Furthermore, we study the issue of feature selection in set-valued decision tables by employing α-dominance-based quantitative rough set models and discuss the relationships between the relative reductions and discernibility matrices. We also present approximate reasoning models based on α-dominance-based quantitative rough sets. Finally, the application of the approach is illustrated by some real-world data sets.