FUZZY LOGIC

Fuzzy Logic

Fuzzy Logic

Question 1 - Type 2 fuzzy logic

a)

Type-2 fuzzy sets and systems generalize (type-1) fuzzy sets and systems so that more uncertainty can be handled. From the very beginning of fuzzy sets, criticism was made about the fact that the membership function of a type-1 fuzzy set has no uncertainty associated with it, something that seems to contradict the word fuzzy, since that word has the connotation of lots of uncertainty.

b)

Let's examine a fuzzy set A. In ordinary set theory, we can do several things with sets. (Think of complements, unions, intersections and such.) We can extend these ideas to fuzzy sets. First, let's examine the complement A¯ of A. The common de?nition for A¯ is that, for all x ? X, we have

µA¯(x) = 1 - µA(x). (2.1)

To de?ne the intersection C = A n B between two sets A and B, we need a t-norm T(a, b) such that µC(x) = T(µA(x), µB(x)) for all x ? X. Such a t-norm must satisfy the following conditions.

T(a, 1) = a, (2.2)

b = c ? T(a, b) = T(a, c), (2.3)

T(a, b) = T(b, a), (2.4)

T(a, T(b, c)) = T(T(a, b), c). (2.5)

The most commonly used t-norms are the standard intersection (also known as the minimum) and the algebraic product, which are respectively de?ned as

T(a, b) = min(a, b) and T(a, b) = ab. (2.6)

The minimum is the largest possible t-norm. To de?ne the union C = A ? B between two sets A and B, we need a t-conorm S(a, b) such that

µC(x) = S(µA(x), µB(x)) for all x ? X.

Such a t-conorm must satisfy the following conditions.

S(a, 0) = a, (2.7)

b = c ? S(a, b) = S(a, c), (2.8)

S(a, b) = S(b, a), (2.9)

S(a, S(b, c)) = S(S(a, b), c). (2.10)

The most commonly used t-conorms are the standard union (also known as the maximum) and the algebraic sum, which are respectively de?ned as

S(a, b) = max(a, b) and S(a, b) = 1 - (1 - a)(1 - b) = a + b - ab. (2.11)

The maximum is the smallest possible t-norm. We can also change fuzzy sets by using hedges. Let's suppose that the fuzzy set A indicates expensive cars. If some element x (say, x = 10,000 euros) has a low membership degree, it is not expensive. But if its membership degree is high, it is expensive. How can we ?nd the set B that indicates very expensive cars or the set C that indicates mildly expensive cars? There are two methods. We can use shifted hedges: we shift the membership function along the domain.

So, µB(x) = µA(x - 5, 000) and µC(x) = µA(x + 3000). We can also use powered hedges: µB(x) = µA(x)2 and µC(x) = p µA(x).

c)

In order to symbolically distinguish between a type-1 fuzzy set and a type-2 fuzzy set, a tilde symbol is put over the symbol for the fuzzy set; so, A denotes a type-1 fuzzy set, whereas à denotes the comparable type-2 ...