Fuzzy Logic in Student Modeling - Advances in Personalized Web-Based Education - page 29

Fig. 2.4 The membership function of a general type-2 fuzzy set
• 1st dimension : the primary variable x (e.g. age, height, grade, temperature)
• 2nd dimension : the primary membership function (PMF), which is a function
and not just a value between 0 and 1.
• 3rd dimension : the secondary membership function (SMF), which is the value
of the membership function at each point on its two-dimensional domain that
is called its footprint of uncertainty (FOU). The value of SMF is, also, range
between 0 and 1.
Using type-2 fuzzy logic can reduce the amount of uncertainty in a system. This is hap-
pened due to the fact that type-2 fuzzy logic offers better capabilities to handle linguis-
tic uncertainties by modeling vagueness and unreliability of information (Liang and
Mendel 2000). Such sets are useful in circumstances where it is difficult to determine
the exact membership function for a fuzzy set, as in modeling a word by a fuzzy set.
When the value of the third dimension is the same (e.g. 1) everywhere, then
the type-2 fuzzy set is called interval type-2 fuzzy set. For an interval type-2 set
the SMF is ignored and only the FOU is used to describe it. The more (less) area
in the FOU the more (less) is the uncertainty (Mendel 2001). The FOU represents
the blurring of a type-1 membership function. It is completely described by its two
bounding functions (Fig. 2.5 ): (i) a lower membership function (LMF) and (ii) an
upper membership function (UMF).
2.2.3 Rule-Based Fuzzy Logic System
Type-2 fuzzy sets are finding very wide applicability in rule-based fuzzy logic
systems (FLSs). The operation of FLSs is based on rules. The rules are expressed
as a collection of IF-THEN statements (e.g. If George's grade at mathematics is
65/100, then he is classified to moderate students). Fuzzy sets are associated with
the terms that appear in the antecedents (IF-part) or consequents (THEN-part) of
rules. For example in the example “if George's grade at mathematics is 65/100,
then he is classified to moderate students”, the fuzzy set 'moderate' appears in