Unit – 4

Introduction to Fuzzy sets

4.1 Crisp set and Fuzzy set

Definition Fuzzy set: A fuzzy set is a combination of the elements having a changing degree of memberships in the set. Here “fuzzy” means vagueness, in other words, the transition among various degrees of the membership complies that the limits of the fuzzy sets are vague and ambiguous. Therefore, the membership of the elements from the universe in the set is measured against a function to identify the uncertainty and ambiguity.

Basis For Comparison:

Definition of Crisp Set:

The crisp set is a collection of objects (say U) having identical properties such as countability and finiteness. A crisp set ‘B’ can be defined as a group of elements over the universal set U, where a random element can be a part of B or not. Which means there are only two possible ways, first is the element could belong to set B or it does not belong to set B. The notation to define the crisp set B containing a group of some elements in U having the same property P, is given below.

Fuzzy systems:

Example 1: Crisp logic (crisp) is the same as Boolean logic (either 0 or 1). Either a statement is true (1) or it is not (0), mean while fuzzy logic captures the degree to which something is true.

Solution: Consider the statement: “The agreed to met at 12’o clock but ben was not punctual.”

Example 2: Crisp sets versus Fuzzy sets

      Characteristic Function

                Short            ave               tall

4.2 Basic concepts of fuzzy sets

Fuzzy sets: Basic concepts

Example 1:

Example 2: A fuzzy set A on R is convex iff A()

For all and all where min denotes the minimum operator.

Solution: (i) Assume that A is convex and let Then and, moreover, for any by the convexity o A. Consequently,

                           A ()

(ii) Assume that a satisfies. We need to prove that for any is convex. Now for any (i.e., A ()and for any

A ()

i.e. .Therefore, is convex for any Hence, A is convex.

Example 3: Consider three fuzzy sets that represent the concepts of a young middle-aged, and old person. The membership functions are defined on the interval [0,80] as follows:

Solution:

4.3 Basic operations on fuzzy sets

Operation on Fuzzy Sets:

Having two fuzzy sets , the universe of information U and an element y of the universe, the following relations express the union, intersection and complement operation on fuzzy sets.

Union/Fuzzy ‘OR’

Let us consider the following representation to understand how the Union/Fuzzy ‘OR’ relation works-

Here represents the ‘max’ operation.

Intersection/Fuzzy ‘AND’

Let us consider the following representation to understand how the Intersection/Fuzzy ‘AND’ relation works-

Here represents the ‘min’ operation.

Complement/Fuzzy ‘NOT’

Let us consider the following representation to understand how the Complement/Fuzzy ‘NOT’ relation works-

4.4 Properties of fuzzy sets:

Let us discuss the different properties of fuzzy sets.

Commutative Property:

Having two fuzzy sets , this property states-

Associative Property:

Having three fuzzy sets this property states-

Distributive Property:

Having three fuzzy sets this property states-

Idempotency Property:

For any fuzzy set this property states-

Identity Property:

For fuzzy set and universal set U, this property states-

Transitive Property:

Having three fuzzy sets , this property states-

If , then

Involution Property:

For any fuzzy set , this property states-

De Morgan’s Law:

This law plays a crucial role in proving tautologies and contradiction. This law states-

Some Solved Examples:

Example 1:

Calculated R (t,h) through min operator

Solution: Now suppose, we want to get information about the humidity when there is the following premise about the temperature:

Temperature is fairly high.

This fact is rewritten as R(t): t is where

Where the fuzzy term is defined as below membership function of in T (temperature)

R(h)= R(t) o R(t,h)

Example 2: Let P = {paris, berlin, Amsterdam} and Q = {Rome, Madrid, Lisbon} be two set of cities, and E = {far, very far, near, very near}. Let R be the fuzzy soft relation over the sets  P and Q  given by (R,E) = {R(for) = {(paris, Rome)/0.60, (paris, madrid/ 0.45, (Paris, Lisbon)/0.40, (Berlin, Rome)/ 0.50, (Berlin, Madrid)/0.65), (Berlin, Lisbon)/0.70,(Amsterdam, Rome)/0.75, (Amsterdam, Madrid)/0.50,(Amsterdam, Lisbon)/0.80}}.This information can be represented in the form of two-dimensional array(matrix) as shown in the Table 4. It is obvious from the matrix given in table 4 that a fuzzy soft relation may be considered as a parameterized fuzzy relation.

Solution:

(1)  Union:

(2)  Intersection:

(3)  Containment:

Example 3: Let U = {} be the set of watches and A = {cheap, costly}, B = {Beautiful, in a golden locket}. Let (F, A) and (G,B) be two soft sets given by F (cheap) = {.The relation is given by the following membership matrices in table 5 and 6.  The membership function of R can also be defined using other appropriate techniques.

Solution:

Reference