The adjective fuzzy is used to describe an entity that has no sharply demarcated boundary. What we understand through natural languages is usually fuzzy.
Examples:
fast car, hot water, cold weather, tall man, effective medicine.
concept of (the Crisp) set is fundamental in mathematics. the crisp set is a classical ordinary set.
It is denoted
by A = { x∈ X |P(x) } For example, the set of points on a unit circle is written as
A = { (x, y) ∈ R *R]X+Y=1}
Here, the universal set is R x R, the set of ordered pairs of real numbers (x, y), the property P(x y) must satisfy is x + y= 1.
Let X be a universal set and [0, 1] be the closed unit interval of real numbers between 0 and 1 including 0 and 1. Then a fuzzy set A on X is a function:
X∈ [0, 1] If X, then A (x) is called the grade of membership of x in A.
α-cut or a-level cut or level cut is one of the most basic concepts in the fuzzy set theory. It connects fuzzy set theory with crisp sets.
Let a e [0, 1]: the crisp set { x ∈ X| A(x) >=α} is called -cut or a-level cut of A and is denotcd by .
α AThus,
strong α- cut: The crisp sct {x∈X| A(x) > α} is called a strict α-cut or strict α-level cut of A and is denoted by α+A. Thus