The notion of a rough set, developed by Pawlak , is an important tool to describe situation of incomplete or partially unknown information. In this article, which is essentially the continuation of , we try to give the characterization of approximation operators in terms of ordinary properties of underlying relations (some of them, as serial and mediate relations, were not available in the Mizar Mathematical Library). Here we drop the classical equivalence- and tolerance-based models of rough sets  trying to formalize some parts of  following also  in some sense (Propositions 1-8, Corr. 1 and 2; the complete description is available in the Mizar script). Our main problem was that informally, there is a direct correspondence between relations and underlying properties, in our approach however , which uses relational structures rather than relations, we had to switch between classical (based on pure set theory) and abstract (using the notion of a structure) parts of the Mizar Mathematical Library. Our next step will be translation of these properties into the pure language of Mizar attributes.
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 Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.
 Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
 Adam Grabowski. Basic properties of rough sets and rough membership function. FormalizedMathematics, 12(1):21-28, 2004.
 Adam Grabowski and Magdalena Jastrzebska. A note on a formal approach to rough operators. In Marcin S. Szczuka and Marzena Kryszkiewicz et al., editors, Rough Setsand Current Trends in Computing - 7th International Conference, RSCTC 2010, Warsaw,Poland, June 28-30, 2010. Proceedings, volume 6086 of Lecture Notes in ComputerScience, pages 307-316. Springer, 2010. doi:10.1007/978-3-642-13529-3 33.
 Artur Korniłowicz. Cartesian products of relations and relational structures. FormalizedMathematics, 6(1):145-152, 1997.
 Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.
 Z. Pawlak. Rough sets. International Journal of Parallel Programming, 11:341-356, 1982. doi:10.1007/BF01001956.
 Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441-444, 1990.
 Andrzej Skowron and Jarosław Stepaniuk. Tolerance approximation spaces. FundamentaInformaticae, 27(2/3):245-253, 1996. doi:10.3233/FI-1996-272311.
 Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1): 115-122, 1990.
 Wojciech A. Trybulec and Grzegorz Bancerek. Kuratowski - Zorn lemma. FormalizedMathematics, 1(2):387-393, 1990.
 Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
 Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.
 Mirosław Wysocki and Agata Darmochwał. Subsets of topological spaces. FormalizedMathematics, 1(1):231-237, 1990.
 Y.Y. Yao. Two views of the theory of rough sets in finite universes. International Journalof Approximate Reasoning, 15(4):291-317, 1996. doi:10.1016/S0888-613X(96)00071-0.
 William Zhu. Generalized rough sets based on relations. Information Sciences, 177: 4997-5011, 2007