Incomplete Information: Rough Set Analysis
The book presents rough set formalisms and methods of modeling and handling incomplete information and motivates their applicability to knowledge representation, knowledge discovery and machine learning. The book focuses on providing representational and
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Studies in Fuzziness and Soft Computing Editor-in-chief Prof. Janusz Kacprzyk Systems Research Institute Polish Academy of Sciences u. Newelska 6 01-447 Warsaw, Poland E-mail: [email protected] Vol. 1. J. Kacprzyk and M. Fedrizzi (Eds.) Fuzzy Regression Analysis, 1992
ISBN 3-7908-0591-2 (ISBN 83-85262-07-5) Vol. 2. A.M. Delgado et aI. (Eds.) Fuzzy Optimization, 1994
ISBN 3-7908-0749-4 Vol. 3. A. Geyer-Schulz Fuzzy Rule-Based Expert Systems and Genetic Machine Learning, 2nd ed. 1996
ISBN 3-7908-0964-0 Vol. 4. T. Onisawa and J. Kacprzyk (Eds.) Reliability and Safety Analyses under Fuzziness, 1995 ISBN 3-7908-0837-7 Vol. 5. P. Bosc and J. Kacprzyk (Eds.) Fuzziness in Database Management Systems, 1995 ISBN 3-7908-0858-X Vol. 6. E. S. Lee and Q. Zhu Fuzzy and Evidence Reasoning, 1995
ISBN 3-7908-0880-6
Vol. 7. B.A. Juliano and
w. Bandler
Tracing Chains-ol-Thought, 1996
ISBN 3-7908-0922-5
Vol. 8. F. Herrera and J. L. Verdegay (Eds.) Genetic Algorithms and Soft Computing, 1996
ISBN 3-7908-0956-X Vol. 9. M. Sato et aI. Fuzzy Clustering Models and Applications, 1997
ISBN 3-7908-1026-6 Vol. 10. L. C. Jain (Ed.) Soft Computing Techniques in Knowledge-based Intelligent Engineering Systems, 1997
ISBN 3-7908-1035-5 Vol. 11. W. Mielczarski (Ed.) Fuzzy Logic Techniques in Power Systems, 1998 ISBN 3-7908-1044-4 Vol. 12. B. Bouchon-Meunier (Ed.) Aggregation and Fusion of Imperfect Information, 1998
ISBN 3-7908-1048-7
Ewa Orlowska (Ed.)
Incomplete Information: Rough Set Analysis With 54 Figures and 23 Tables
Springer-Verlag Berlin Heidelberg GmbH
Prof. Dr. Ewa Orlowska Institute of Telecommunications ul. Szachowa 1 PL-04-894 Warsaw, Poland
ISBN 978-3-7908-2457-5 Library of Congress Cataloging-in-Publication Data Die Deutsche Bibliothek - CIP-Einheitsaufnahme Incomplete infonnation: rough set analysis; with 23 tables / Ewa Orlowska (ed.). (Studies in fuzziness and soft computing; Vol. 13) ISBN 978-3-7908-2457-5 ISBN 978-3-7908-1888-8 (eBook) DOI 10.1 007/978-3-7908-1888-8 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is pennitted only under the provisions of the Gennan Copyright Law of September 9, 1965, in its current version, and pennission for use must always be obtained from Springer-Verlag Berlin Heidelberg GmbH. Violations are liable for prosecution under the Gennan Copyright Law. © Springer-Verlag Berlin Heidelberg 1998 Originally published by Physica-Verlag Heidelberg New York in 1998 The use of general descriptive names, registered names, uau"lIlaf1 0 ((x;y)* .y-l) ·x** > 0 ((x;y)* .y-l)·x > 0 (x; y) . (x; y)*
>0
,
o
a contradiction.
Just like relation algebras, R2 As have very strong structural properties: An algebra A is a discriminator algebra if there is some term operation f in the language of A such
