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:
Basis for comparison  Fuzzy Set  Crisp Set 
Basic  Prescribed by vague or ambiguous properties.  Defined b precise and certain characteristics. 
Property  Elements are allowed to be partially included in the set.  Element is either the member of a set or not.

Application  Used in fuzzy controllers  Digital design 
Logic  InfiniteValued  Bivalued 
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
Crisp set Short average under 170cm tall 170 to 180cm over 180cm 
Characteristic Function
Short ave tall
B  A  C 
 short  ave  tall 
A  0  1  0 
B  1  0  0 
C  0  0  1 
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 middleaged, and old person. The membership functions are defined on the interval [0,80] as follows:
Solution:
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
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:
t  20  30  40 
0.1  0.5  0.9 
h  20  50  70  90 
0.2  0.6  0.7  1 
 20  50  70  90 
20  0.1  0.1  0.1  0.1 
30  0.2  0.5  0.5  0.5 
40  0.2  0.6  0.7  0.9 
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)
t  20  30  40 
0.01  0.25  0.81 
R(h)= R(t) o R(t,h)
 20  50  70  90 
20  0.1  0.1  0.1  0.1 
30  0.2  0.5  0.5  0.5 
40  0.2  0.6  0.7  0.9 
t  20  30  40 
0.01  0.25  0.81 
h  20  50  70  90 
0.2  0.6  0.7  0.81 
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 twodimensional 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:
R(far)  Rome  Madrid  Lisbon 
Paris  0.60  0.45  0.40 
Berlin  0.55  0.65  0.70 
Amsterdam  0.75  0.50  0.80 
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:
R(Costly beautiful)  
0.65  1  0.80  0.70  0.80  0.75  
0.49  0.75  0.60  0.53  0.60  0.56  
0.52  0.80  0.64  0.56  0.64  0.60  
0.35  0.55  0.44  0.39  0.55  0.42  
0.59  0.90  0.72  0.63  0.72  0.68  
0.53  0.85  0.68  0.60  0.68  0.64 
Reference