normal fuzzy set and a subnormal fuzzy set

In fuzzy logic, a normal fuzzy set and a subnormal fuzzy set are two different types of fuzzy sets that can be distinguished based on their membership functions and the extent to which they cover the universe of discourse.

1. Normal Fuzzy Set:

A normal fuzzy set is a fuzzy set whose membership function has a maximum value of 1, indicating full membership, for at least one element in the universe of discourse. In other words, there exists an element in the set for which the membership value is 1.

Example:

Let's consider a normal fuzzy set A defined on the universe of discourse U = {1, 2, 3, 4, 5}. The membership function of A is defined as follows:

μA(1) = 0.4

μA(2) = 0.8

μA(3) = 1.0

μA(4) = 0.6

μA(5) = 0.2

In this example, the membership function of A has a maximum value of 1 for the element 3. Therefore, fuzzy set A is a normal fuzzy set.

2. Subnormal Fuzzy Set:

A subnormal fuzzy set is a fuzzy set whose membership function does not have a maximum value of 1 for any element in the universe of discourse. In other words, the membership values of all elements in the set are less than 1.

Example:

Consider a subnormal fuzzy set B defined on the universe of discourse U = {1, 2, 3, 4, 5}. The membership function of B is defined as follows:

μB(1) = 0.2

μB(2) = 0.4

μB(3) = 0.6

μB(4) = 0.4

μB(5) = 0.2

In this example, none of the membership values in the membership function of B are equal to 1. Therefore, fuzzy set B is a subnormal fuzzy set.

To summarize, a normal fuzzy set has at least one element with a membership value of 1, while a subnormal fuzzy set does not have any element with a membership value of 1.