Rough Sets pp 465-500

# From Rough to Fuzzy

• Lech Polkowski
Part of the Advances in Soft Computing book series (AINSC, volume 15)

## Abstract

We have witnessed the development of logical calculi with truth values ranging continuously from 0 to 1 and in the development of the fuzzy sentential logic we have treated the set of logical axioms as a fuzzy set. The fuzzy sentential calculus in Chapter 13 has been developed in the framework of residuated lattices and an essential usage has been made of the adjoint pair (⊗, →). In the wider perspective of fuzzy calculi on sets it has turned useful to extend the notion of an adjoint pair to the notion of a pair (T,→ T ) where T is a triangular norm (or, tnorm) and → T is the induced residuated implication. t — norms and duals of them, tconorms, may be applied in the development of algebra of fuzzy sets viz. t — norms determine intersections of fuzzy sets according to the formula X A∩B (x) = T ( XA (x), XB (x)) where T is a t — norm and XA is the fuzzy characteristic (membership) function of the fuzzy set A while t — conorms may be used in determining unions of fuzzy sets via X A∪B (x) = T ( XA (x), XB (x)) where S is a t — conorm.

## Keywords

Residuated Lattice Fuzzy Partition Triangular Norm Adjoint Pair Fuzzy Similarity
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. [Aczél49]
J. Aczél, Sur les operations defines pour les nombres réels, Bull. Soc. Math. France, 76 (1949), pp. 59 — 64.Google Scholar
2. [Arnold63]
V. I. Arnold, On functions of three variables, Amer. Math. Soc. Transi., 28 (1963), pp. 51 — 54.Google Scholar
3. [Caianiello87]
E. R. Caianiello, C — calculus: an overview, in: E.R. Caianiello and M. A. Aizerman (eds.), Topics in the General Theory of Structures,Reidel, Dordrecht, 1987.
4. [Cattaneo98]
G. Cattaneo, Abstract approximation spaces for rough theories, in: L. Polkowski and A. Skowron (eds.), Rough Sets in Knowledge Discovery. Methodology and Applications, vol. 18 in Studies in Fuzziness and Soft Computing, Physica Verlag, Heidelberg, 1998, pp. 59 — 98.Google Scholar
5. [Cattaneo97]
G. Cattaneo, Generalized rough sets. Preclusivity fuzzy — intuitionistic (BZ) lattices, Studia Logica, 58 (1997), pp. 47–77.
6. [Cattaneo — Nistico89]
G. Cattaneo and G. Nisticb, Brouwer — Zadeh posets and three valued Lukasiewicz posets, Fuzzy Sets Syst., 33 (1989), pp. 165 —190.Google Scholar
7. [Dempster67]
A. P. Dempster, Upper and lower probabilities induced by a multiple — valued mapping, Annals Math. Stat., 38 (1967), pp. 325 — 339.Google Scholar
8. [Driankov93]
D. Driankov, H. Hellendoorn, and M. Reinfrank, An Introduction to Fuzzy Control, Springer Verlag, Berlin, 1993.
D. Dubois and H. Prade, Putting rough sets and fuzzy sets together, in: R. Slowinski (ed.),Intelligent Decision Support. Handbook of Applications and Advances of the Rough Sets Theory, Kluwer, Dordrecht, 1992, pp. 203 — 232.Google Scholar
D. Dubois and H. Prade (with coll.), Possibility Theory: An Approach to Computerized processing of Uncertainty, Plenum Press, New York, 1988.Google Scholar
11. [Farinas del Cerro86]
L. Farinas del Cerro and H. Prade, Rough sets, twofold fuzzy sets and modal logic — fuzziness in indiscernibility and partial information, in: A. Di Nola and A. G. S. Ventre (eds.), The Mathematics of Fuzzy Systems, Verlag TUV Rheinland, Köln, 1986.Google Scholar
12. [Faucett55]
W. M. Faucett, Compact semigroups irreducibly connected between two idempotents, Proc. Amer. Math. Soc., 6 (1955), pp. 741 — 747.Google Scholar
13. [Höhle88]
U. Höhle, Quotients with respect to similarity relations,Fuzzy Sets Syst., 27 (1988), pp. 31 — 44.Google Scholar
14. [Kolmogorov63]
A. N. Kolmogorov, On the representation of continuous functions of many variables by superposition of continuous functions of one variable and addition,Amer. Math. Soc. Transi., 28 (1963), pp. 55 — 59.Google Scholar
15. [Lindenbaum33]
A. Lindenbaum, Sur les ensembles dans lesquels toutes les équations d’une famille donnée ont un nombre de solution fixé davance, Fund. Math., 20 (1933), p. 20.Google Scholar
16. [Ling65]
C. — H. Ling, Representation of associative functions, Publ. Math. Debrecen, 12 (1965), pp. 189–212.Google Scholar
17. [Mantaras–Valverde88]
L. de Mântaras and L. Valverde, New results in fuzzy clustering based on the concept of indistinguishability relation, IEEE Trans. on Pattern Analysis and Machine intelligence, 10 (1988), pp. 754–757.
18. [Menger42]
K. Menger, Statistical metrics, Proc. Nat. Acad. Sci. USA, 28 (1942), pp. 535–537.
J. Menu and J. Pavelka, A note on tensor products on the unit interval, 17 (1976), pp. 71–83.
20. [Mostert–Shields57]
P. S. Mostert and A. L. Shields, On the structure of semigroups on a compact manifold with boundary, Ann. Math., 65 (1957), pp. 117–143.
21. [Nakamura88]
A. Nakamura, Fuzzy rough sets, Notes on Multiple–Valued Logic in Japan, 9 (1988), pp. 1–8.Google Scholar
22. [Nakamura–Gao91]
A. Nakamura and J. M. Gao, A logic for fuzzy data analysis, Fuzzy Sets Syst., 39 (1991), pp. 127–132.
23. [Pal–Skowron99]
S. K. Pal and A. Skowron, Rough–Fuzzy Hybridization. A New Trend in Decision–Making, Springer Verlag, Singapore, 1999.
24. [Pavelka79a,b,c]
J. Pavelka, On fuzzy logic I, II, III,Zeit. Math. Logik Grund. Math., 25 (1979), pp. 45-52, 119-134, 447-464.Google Scholar
25. [Pawlak85c]
Z. Pawlak, Rough sets and fuzzy sets, Fuzzy Sets Syst., 17 (1985), pp. 99–102.
26. [Pedrycz99]
W. Pedrycz, Shadowed sets: bringing fuzzy and rough sets,in: [Pal–Skowron], pp. 179-199.Google Scholar
27. [Ruspini9l]
E. H. Ruspini, On the semantics of fuzzy logic, Int. J. Approx. Reasoning, 5 (1991), pp. 45–88.
28. [Schweizer–Sklar83]
B. Schweizer and A. Sklar, Probabilistic Metric Spaces, North–Holland, Amsterdam, 1983.
29. [Shafer76]
G. Shafer, A Mathematical Theory of Evidence, Princeton U. Press, Princeton N. J., 1976.Google Scholar
30. [Sierpiński34]
W. Sierpiński, Remarques sur les fonctions de plusieurs variables réelles,Prace Matematyczno — Fizyczne, 41 (1934), pp. 171 — 175.Google Scholar
31. [Valverde85]
L. Valverde, On the structure of F–indistinguishability operators,Fuzzy Sets Syst., 17 (1985), pp. 313 — 328.Google Scholar
L. A. Zadeh, Fuzzy sets as a basis for the theory of possibility,Fuzzy Sets Syst., 1 (1978), pp. 3 — 28.Google Scholar
L. A. Zadeh, Similarity relations and fuzzy orderings, Information Sciences, 3 (1971), pp. 177–200.
L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), pp. 338–353.