Rough set theory provides a systematical tool for attribute reduction and rule induction in information systems. There is much need to use sophisticated theories or tools to make it more adaptive to practical problems. This paper constructs an isomorphism from equivalence relations to 2-circuit matroids, and provides two equivalent characterizations of attribute reductions of information systems with matroidal approaches. Firstly, we establish a matroidal structure of rough sets. Specifically, an approach to inducing a matroid fromrnan equivalence relation is proposed. Several equivalent formulations of independent sets of the matroid by an equivalence relation are presented. Secondly, an approach is provided to induce an equivalence relation from a matroid. Thirdly, an isomorphism from equivalence relations to 2-circuit matroids is constructed. We also use matroidal approaches to studying rough sets；specially, it is proved that the upper approximation operator with respect to an equivalence relation is equal to the closure operator of the matroid induced by the equivalence relation. Finally, attribute reductions of information systems are equivalently formulated with rank functions and closure operators of matroids induced by attributes.
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