Unit – 4

Introduction to Fuzzy sets

Q1) Crisp logic (crisp) is the same as Boolean logic (either 0 or 1). Either a statement is true (1) or it is not (0), meanwhile fuzzy logic captures the degree to which something is true.

A1)

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

Q2) Crisp sets versus Fuzzy sets

A2)

Characteristic Function

                Short            ave               tall

Q3)

A3)

Q4) A fuzzy set A on R is convex iff A()

For all and all where min denotes the minimum operator.

A4)

(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.

Q5) 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:

A5)

Q6)

Calculated R (t,h) through min operator

A6)

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)

Q 7: 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.

A7)

(1)  Union:

(2)  Intersection:

(3)  Containment:

Q8) 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.

A8)