# Recent Trends and Applications of Fuzzy Logic

Pankaj Kumar Srivastava (Jaypee Institute of Information Technology, India) and Dinesh C. S. Bisht (Jaypee Institute of Information Technology, India)
DOI: 10.4018/978-1-5225-5709-8.ch015
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## Abstract

Classical set theory, which is based on dichotomy, is not applicable in cases where vagueness is involved. Fuzzy logic is based on the idea of relative graded membership. Fuzzy logic has all the strength to cope with vagueness, uncertainty, and imprecision. Fuzzy logic is a tool that connects human cognitive relations to computers, since computers are not at all good in reading imprecise and vague data. Fuzzy logic is gaining its popularity in various field of research. It found its application in decision making, identification, time series, pattern recognition, optimization, and control. This chapter discusses fuzzy logic, fuzzy sets, and major applications of fuzzy computing.
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## Classical Sets Vs. Fuzzy Sets

### Definition of Classical (Crisp) Set

A well-defined collection of objects, from the same individuality, is called classical set commonly as set. Elements of the set A are denoted by x. Some general examples of classical sets include set of integers, set of real numbers, set of complex numbers etc.

### Cardinality of Classical Set

The number of elements in a set is called cardinal number of the set. Denoted by where A is any given set. If number of members in a set is finite then cardinal number of that set is finite otherwise set will have infinite cardinal number. Examples given in the dentition of classical set have infinite cardinal number.

### Subset

Subset is a group of members from a set. For example, a set A is a subset of universal set U.

### Power Set

A power set of a set A is defined as all feasible collection of subsets of A. It is denoted by P (A). The number of elements in power set indicates the no of possible subsets of A, known as cardinality of P (A). Cardinality of P (A) is given by.

### Operations in Classical Sets

#### Union of Two Classical Sets

Union of two classical set A and B is a set which contains members of both the sets. Mathematically denoted by and defined as where A and B are two classical sets defined on a universal set E.

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