STUDY OF CONVEXITY IN SETS AND FUZZY SETS
Abstract
In this paper, we present an overview of both classical and fuzzy convexity, particularly, in conjunction with continuity5 and some topological concepts, and provide proofs of some of their algebraic properties along with suitable illustrations. We have extended the work of (Ammar E. E., 1999) by proving that if and are two convex fuzzy sets, then , , , and are also convex fuzzy sets. We have also shown that the convexity of a fuzzy set implies semistrict quasi-convexity
References
Ammar, E. E. (1999). Some Properties of Convex Fuzzy Sets and Convex Fuzzy Cones. Fuzzy Sets and Systems, 106, 381-386.
Ammar, E. E., & Metz, J. (1992). On fuzzy convexity and parametric fuzzy optimization. Fuzzy Sets and Systems, 49, 135-142.
Brown, J. (1971). A Note on Fuzzy Sets. Journal of information and control, 18, 23-39.
Cheng, J., Syau, Y. R., & Ting, C. J. (2004). Convexity and semi continuity of fuzzy sets. fuzzy sets and systems, 143, 459-469.
Drewniak, J. (1987). Convex and Strongly Convex Fuzzy Sets. Journal of Mathematical Analysis and Applications, 126, 292 – 300.
Katsaras, A. K., & Liu, D. B. (1977). Fuzzy Vector Spaces and Fuzzy Topological Vector Spaces. Journal of Mathematical Analysis and Applications, 58, 135-146.
Liu, Y. M. (1985). Some Properties of Convex Fuzzy Sets. Journal of Mathematical Analysis and Applications, 111, 119-129.
Lowen, R. (1980). Convex fuzzy sets. Fuzzy Sets and Systems, 3, 291–310.
Syau, Y. R. (2000). Closed and Convex Fuzzy Sets. Fuzzy Sets and Systems, 110, 287-291.
Syau, Y. R., Lee, E. S., & Lixing, J. L. (2004). Convexity and Upper Semi Continuity of Fuzzy Sets. Journal of Computers and Mathematics with Applications, 48, 117-129.
Wu, S. Y., & Cheng, W. (2004). A note on fuzzy convexity. Applied Mathematics Letter, 17, 1127-1133.
Wu, S. Y., & Cheng, W. H. (2004). A note on fuzzy convexity. Applied Mathematics Letter, 17, 1127-1133.
Yang, X. (1993). A Note on Convex Fuzzy Set. Fuzzy Sets and Systems, 53, 117 118.
Yang, X. (1995). Some Properties of Convex Fuzzy Set. Fuzzy Sets and Systems, 72, 129-132.
Zadeh, L. A. (1965). Fuzzy sets. Information and Control,, 8, 338-353.
Zadeh, L. A. (1975). The concept of linguistic variable and its application to approximate reasoning. Information Science, 8, 199-249.
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