Fuzzy Sets And Their Applications By Pundir Pdf Free Download

The concept of fuzzy set was developed in 1965. Since then, researchers have used this key set in many disciplines. As a brand-new conceptual system to support human-centric framework, fuzzy set has proved quite promising and effective in modeling human involvement in human-based intelligence to attain modernity in many departments like data analyzing, data mining, image coding and explaining, as well as in intelligence systems. Fuzzy set has also become an acknowledged research subject in both pure as well as in applied mathematics and statistics, showing how this theory is highly applicable and productive in many applications. Despite being a core subject for many years, fuzzy set still attracts researchers for putting forth solutions for prime issues with certain features questioned by these notions. Fuzzy set can effectively deal with a wide spectrum of problems of the physical world via cooperation, which may be beyond the capability of classical techniques. This means the fuzzy set could have the ability to handle a wide range of problems, for instance, decision making, intelligent data analysis, processing information, pattern recognition, and optimization.

The goal of this Special Issue is to dive deeper into the new trends of fuzzy set theory and the extension of fuzzy set theory with applications in group theory, ring theory, statistics, topological spaces, graph theory, decision making and other engineering applications.

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Because of the constraints in (1), all points must completely allocate their memberships to each cluster [19]. The fuzzy weight centre of gravity of the data is used to define the cluster centre (centroid)X.

ELM uses arbitrary hidden node parameters and the tune-free trained approach to FFNN instead of iteratively upgrading network parameters as in traditional gradient descent algorithms. ELM is flexible because it employs a hidden activation function, as demonstrated by the universal approximate ability theorem. Almost any nonlinear piecewise continuous function and its linear combination perform well in the ELM algorithm [22]. The extreme learning machine (ELM) is a fast convergent training method for single hidden layer feedforward neural networks (SLFNs). This type of SLFN allows for faster convergence training and avoids the need for many iterations to update the hidden layer weights. Compared to other classical learning algorithms in applications with increasing noise, ELM appears to outperform ELM in regression and classification tests. With a single hidden layer of neurons and random feature mapping, an ELM model learns quicker than other models. High dimensions and large data sets have aroused substantial scholarly interest in the low computing complexity.

14 Unit M.Sc. Paper : Fuzzy Sets and Their Application- MM : 35 dea of fuzzy set and membership function, Definition of a fuzzy set, membership function, representation of membership function, General definitions and properties of fuzzy sets, Support, height, equality of two fuzzy sets, containment, examples. Union and ntersection of two fuzzy sets, Complement of a fuzzy set, normal fuzzy set, -cut set of a fuzzy set, strong -cut, convex fuzzy set, a necessary and sufficient condition for convexity of a fuzzy set (Theorem 1), Decomposition of fuzzy sets, Degree of sub sethood, Level set of a fuzzy set, Cardinality, fuzzy cardinality, examples. Other important operations on fuzzy sets, Product of two fuzzy sets, Product of a fuzzy set with a crisp number, Power of a fuzzy set, Difference of two fuzzy sets, Disjunctive sum of two fuzzy sets, example. General properties of operations on fuzzy sets, Commutativity, associativity, distributivity, dempotent law, identities for operations, Transitivity, involution, Demorgans laws, proofs and examples, Some important theorems on fuzzy sets, set inclusion of fuzzy sets and corresponding -cuts and strong -cuts (Theorem 1). Comparison of -cut and strong -cut, Order relation of scalars is inversely preserved by set inclusion of corresponding -cuts and strong -cuts, -cut of union and intersection of two fuzzy sets, -cut of complement of a fuzzy set (Theorem 2), Examples, -cuts and strong -cuts of union and intersection of arbitrary collection of fuzzy sets. Text Books : 1 Fuzzy Sets and their Applications by Pundir and Pundir, Pragati Prakashan (PP 30-6). Reference Books : 1. Fuzzy sets and Fuzzy Logic by G.J. Klir and B. Yuan, Prentice Hall of ndia, New Delhi, Fuzzy set Theory and its Applications by H.J. Zimmermann, Allied publishers Ltd, New Delhi 1991.

20 Unit M.Sc. Paper : Fuzzy Sets and their Applications - Fuzzy sets: Basic Definitions, -level sets, Convex fuzzy set, Basic operations on fuzzy sets, types of fuzzy sets, Extensions: Types of fuzzy sets, Further operations on fuzzy sets, Cartesian product, Algebraic products, Bounded sum and Difference, t-norm & t-conorm. Extension principle and applications, Zadeh extension principle, image and inverse image of fuzzy sets, fuzzy numbers, algebraic operations with fuzzy numbers, extended operation and its properties, Special extended operation, addition, subtraction, product and division of fuzzy numbers. Fuzzy relations on fuzzy sets, The union & intersection of fuzzy relations, Composition of fuzzy relations, max-* and max-product compositions, minmax composition and its properties, reflexivity, symmetry, transitivity, and their examples, special fuzzy relations, similarity relation. Fuzzy graphs: Definition and Examples, Fuzzy sub-graph, Spanning subgraph, path in a fuzzy graph, strength and length of a path, -length and - distances, connected nodes, fuzzy forest, fuzzy tree, Examples, Fuzzy Analysis: Fuzzy functions on fuzzy sets, classical function, fuzzy function, Examples. Fuzzy Logic; An overview of classical logic, ts connectives, Tautologies, Contradiction, Fuzzy logic, logical connectives for fuzzy logic, Examples, Approximate reasoning, its rules, examples, other of implication operations, Linguistic hedges, Fuzzy quantifiers, Examples. MM : 35 Text Books : 1. Fuzzy set theory and its Applications by H.J. Zimmermann, Allied Publishers Ltd., New Delhi, 1991 (For Units to ). 2. Fuzzy Logic with Engineering Applications by T.J. Ross, John Wiley & Sons, nd Ed., 2005 (For Unit ). Reference Book : 1. Fuzzy sets and Fuzzy Logic Theory & Application by G.J. Klir and B. Yuan, Prentice Hall of ndia, New Delhi, (2000).

APPLIED MATHEMATICS ADVANCED LEVEL INTRODUCTION This syllabus serves to examine candidates knowledge and skills in introductory mathematical and statistical methods, and their applications. For applications e24fc04721