IMAT5119 Critically evaluate fuzzy logic approaches to solve computational problems exhibiting uncertainty and imprecision.

Assignment - Theoretical

Learning outcome 1: Critically evaluate fuzzy logic approaches to solve computational problems exhibiting uncertainty and imprecision.

Learning outcome 2: Have a comprehensive understanding of the successful application of fuzzy logic to several problem domains and be capable of judging whether the fuzzy paradigm might be fruitful in a novel situation.

Question 1 - Fuzzy sets

A fuzzy set A in X (classical set of objects, called the universe, whose generic elements are denoted x) is a set of ordered pairs:

A={(x,?_A (x))|x?X,?_A (x)?[0,1]}.

The ?-cut (??[0,1]) of a fuzzy set A is the ordinary set

A_?={x?X|?_A (x)??}.

The strong ?-cut (??[0,1]) of a fuzzy set A is the ordinary set

A_(?^+ )={x?X|?_A (x)>?}.

The set of all distinct numbers in [0, 1] that are employed as membership grades of the elements of X in A is called the level set of A, denoted by L(A).

Assume minimum and maximum operators for the intersection and union of fuzzy sets. Answer the following:

Given any two fuzzy sets A and B, prove that the following properties hold:

(A?B)_?=A_??B_? and (A ?B)_?=A_??B_?.

The support of A, denoted supp(A), is defined as the set of elements of X that have nonzero membership in A. The core of A, denoted core(A), is defined as the set of elements of X that have membership in A equal to 1.

How do supp(A) and core(A) relate to the ?-cuts and the strong ?-cuts of A?

Given the discrete fuzzy sets A={(0.2,x_1 ),(0.4,x_2 ),(0.6,x_3 ),(0.8,x_4 ),(1,x_5)}, obtain L(A), and provide all the distinct ?-cuts of A.

What is the relationship between A_(?_1 ) and A_(?_2 ) when ?_1< ?_2?

The membership function of A can be expressed in terms of the characteristic functions of its ?-cuts according to the formula:

?_A (x)=sup-(??[0,1] ) ???_(A_? ) (x) where ?_(A_? ) (x)={(1 iff x?A_?@0 otherwise)

sup means superior (the maximum value of those obtained when multiplying the distinct values of ? in the level set L(A) with 1 or 0 - the value of ?_(A_? ) (x) - depending on whether an x value belongs or not to the alpha-level set A_?).

In the case of a discrete fuzzy set we have ??L(A). Show that this is true for the discrete fuzzy set given in c).