Fuzzy Logic in Student Modeling - Advances in Personalized Web-Based Education - page 28

height; S = {1, 2, 3, …, 120} for age; S = {0, 1, 2, …, 100} for grades) and x ∈
S. In other words, x is a particular value that belongs to the set S. A fuzzy set FS is
a pair (x, μ (x)), where x ∈ S and μ (x): S ₒ [0, 1]. In other words, for each x ∈ S,
there is a value μ (x) between 0 and 1, which declares the membership degree of x
to the fuzzy set FS.
• If μ (x) = 0, then x is not included in FS
• If μ (x) = 1, then x is fully included in FS
• If 0 < μ (x) < 1, then x is partially included in FS
2.2.1 Type-1 Fuzzy Sets
This first approach of fuzzy sets theory, which points that the value of the mem-
bership function of a fuzzy set can range between 0 and 1, is called type-1 fuzzy
sets. Two common examples of a membership function of type-1 fuzzy sets are
depicted in Fig. 2.3 . Type-1 fuzzy sets have been criticized about their ability to
handle uncertainty. It has been advocated that it is not reasonable to use an accu-
rate membership function for something uncertain. Type-1 fuzzy sets used in con-
ventional fuzzy systems cannot fully handle the uncertainties that are present in
intelligent systems (Castillo and Melin 2008). To handle these uncertainties, Lotfi
Zadeh (1975) proposed a more sophisticated kind of fuzzy sets theory that is
called type-2 fuzzy sets (Mizumoto and Tanaka 1976; Mendel 2001).
2.2.2 Interval Type-2 Fuzzy Sets
The concept of a type-2 fuzzy set was introduced first by Zadeh (1975) as an extension
of the type-1 fuzzy set. In particular, the membership function of a general type-2 fuzzy
set is three-dimensional (Fig. 2.4 ):
Fig. 2.3 Examples of type-1 fuzzy sets