Chapter 2: Mathematical Representation and Operations in Fuzzy Sets



Chapter 2: Mathematical Representation and Operations

2.1 Introduction

Fuzzy sets extend the concept of classical sets by allowing gradual assessment of the membership of elements in a set. Instead of having only two membership values (0 or 1) as in classical set theory, fuzzy sets allow any real number between 0 and 1, enabling a better representation of uncertainty and vagueness. This chapter elaborates on how fuzzy sets are mathematically represented and the various operations that can be performed on them.


2.2 Mathematical Representation of Fuzzy Sets

A fuzzy set A in a universe of discourse X is characterized by a membership function:

Each element x in X is mapped to a membership value that indicates the degree of membership of x in the fuzzy set A.

Example:

Let X = {1, 2, 3, 4, 5}

Define a fuzzy set A as:

This means the element 1 belongs to A with a membership value of 0.1, and so on.


2.3 Basic Operations on Fuzzy Sets

Fuzzy set operations are extensions of classical set operations, but the logic is applied over the membership functions.

2.3.1 Union (OR Operation)

For two fuzzy sets A and B over the universe X, the union is defined as:

Example:

Let
A = {(1, 0.2), (2, 0.5), (3, 0.7)}
B = {(1, 0.6), (2, 0.3), (3, 0.9)}

Then:


2.3.2 Intersection (AND Operation)

Example:

Using the same sets A and B:


2.3.3 Complement (NOT Operation)

The complement of a fuzzy set A is given by:

Example:

If A = {(1, 0.2), (2, 0.5), (3, 0.7)}, then:


2.4 Other Fuzzy Set Operators

In addition to the basic operations, there are several other important fuzzy set operators:

2.4.1 Algebraic Sum

2.4.2 Algebraic Product

2.4.3 Bounded Sum

2.4.4 Bounded Difference

2.4.5 Drastic Union

2.4.6 Drastic Intersection


2.5 Properties of Fuzzy Set Operations

Property Operation
Commutativity ,
Associativity
Distributivity
Idempotency ,
Law of Excluded Middle (not necessarily full membership)
Law of Contradiction (some overlap may exist)

2.6 Graphical Representation

Graphically, fuzzy sets are represented using curves over the universe of discourse. The x-axis represents the elements of the universe, and the y-axis shows the degree of membership (from 0 to 1). Plotting these helps in visualizing operations like union (taking the upper envelope of curves), intersection (taking the lower envelope), and complement (flipping the curve over the horizontal line ).


2.7 Conclusion

This chapter presented the foundational operations on fuzzy sets, including union, intersection, complement, and other operators like algebraic sum, bounded difference, and drastic operations. These mathematical tools are essential for handling uncertainty and are widely used in fuzzy logic systems, decision making, and fuzzy control applications.


2.8 Exercises

1. Define a fuzzy set for the linguistic variable “Young Age” over the universe {10, 15, 20, 25, 30, 35}. Assign suitable membership values.
2. Given the following fuzzy sets:
A = {(1, 0.2), (2, 0.6), (3, 0.9)}
B = {(1, 0.5), (2, 0.4), (3, 0.7)}
Compute:
a)
b)
c)

3. Use algebraic sum and algebraic product to compute new fuzzy sets from question 2.
4. Discuss why the Law of Excluded Middle does not strictly apply in fuzzy set theory.
5. Draw the graphs of a fuzzy set and its complement.

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