operations of fuzzy set
This page provides a comprehensive definition of fuzzy set and explores the core operations of fuzzy set. By studying a fuzzy set with example, you will understand the membership function of fuzzy set and how the definition of fuzzy set applies to uncertain data modeling.
What is a Fuzzy Set?
In classical set theory (Crisp Sets), an element either belongs to a set or it does not. Membership is binary: 0 or 1. However, Fuzzy Sets, introduced by Lotfi A. Zadeh in 1965, allow for partial membership. This means an element can belong to a set with a “degree of membership” ranging anywhere from 0 to 1.
Definition
A fuzzy set A in a universe of discourse U is defined by a membership function μA(x) which associates each element x in U with a real number in the interval [0, 1].
Operations of Fuzzy Sets
The standard operations for fuzzy sets are generalizations of classical set operations:
- Union (OR): The membership is the maximum of the two.
μA∪B(x) = max(μA(x), μB(x)) - Intersection (AND): The membership is the minimum of the two.
μA∩B(x) = min(μA(x), μB(x)) - Complement (NOT): The membership is the subtraction from 1.
μA’(x) = 1 – μA(x)
Let A = { (x1, 0.3), (x2, 0.7) }
Let B = { (x1, 0.5), (x2, 0.4) }
Union (A ∪ B): { (x1, 0.5), (x2, 0.7) } (Taking max values)
Intersection (A ∩ B): { (x1, 0.3), (x2, 0.4) } (Taking min values)
What is a Fuzzy Number?
A Fuzzy Number is a special type of fuzzy set where the membership function is convex and normal, and the universe of discourse is the set of real numbers. It represents an interval of values rather than a single crisp point.
Triangular Fuzzy Number (TFN)
The most common type is the Triangular Fuzzy Number, represented as (a, b, c), where:
- a: Lower bound (Membership = 0)
- b: Peak/Core (Membership = 1)
- c: Upper bound (Membership = 0)
Operations of Fuzzy Numbers
Given two triangular fuzzy numbers A = (a1, a2, a3) and B = (b1, b2, b3), the basic arithmetic operations are:
| Operation | Formula |
|---|---|
| Addition | (a1+b1, a2+b2, a3+b3) |
| Subtraction | (a1-b3, a2-b2, a3-b1) |
A = (2, 4, 6) and B = (1, 3, 5)
A + B: (2+1, 4+3, 6+5) = (3, 7, 11)
A – B: (2-5, 4-3, 6-1) = (-3, 1, 5)
Designed by: Dr. M.U. Mirza
Mathematical Researcher & Educator