Fuzzy Sets
Crisp sets are useful but problematic in many situations. For instance, let’s
examine the universe of discourse of all IQs, and let’s define sets for
Dumb, Average, and Clever like so:
Dumb = {70, 71, 72, … 89}
Average = {90, 91, 92, … 109}
Clever = {110, 111, 112, … 129}
A graphical way of showing these crisp sets is shown in Figure 10.3. Note
how the degree of membership of an element in any of the sets can be
either 1 or 0.
People’s intelligence can now be categorized by assigning them to one of
these sets based upon their IQ score. Clearly though, a person with an IQ of
109 is well above average intelligence and probably the majority of his
peers would categorize him as clever. He’s certainly much more intelligent
than a person who has a score of 92 even though both fall into the same
category. It’s also ridiculous to compare a person with an IQ of 79 and a
person of IQ 80 and come to the conclusion that one is dumb and the other
isn’t! This is where crisp sets fall down. Fuzzy sets allow elements to be
assigned to them to a matter of degree.
Defining Fuzzy Boundaries with Membership Functions
A fuzzy set is defined by a membership function. These functions can be
any arbitrary shape but are typically triangular or trapezoidal. Figure 10.4
shows a few examples of membership functions. Notice how they define a
gradual transition from regions completely outside the set to regions com

pletely within the set, thereby enabling a value to have partial membership
to a set. This is the essence of fuzzy logic.
Fuzzy Logic  419
Fuzzy Sets
Figure 10.3