Chapter X: Introduction to Fuzzy Set Theory Uncertainty is universal ...

4 0.5 0.5 0.5 2.0 1.0
Table X.2 Risk output for various smoking habits for w 1 =1.0, w 2 =10.0, w 3 =1.0, w 4 =0.5 and for the input
values stated in the text.
Smoking 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Risk 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.0 1.0 1.0
Additionally, for a given choice of parameters, a “what-if” game can be played. In the above example,
with the parameters as in case 2 of Table X.1, we can examine the change in cancer risk given by
changing a patient’s smoking habits, as displayed in Table X.2.
x.xx Alpha-cuts, the Decomposition Theorem, and the Extension Principle
Suppose you want to take the average, or weighted average of a bunch of senor measurements that are
uncertain. We might model the uncertainty by fuzzy numbers, i.e., by normal, convex fuzzy sets over the
real line. You know what normal fuzzy sets are; intuitively convex fuzzy subsets of the real numbers
have membership function that “go up” for awhile and then “go down”, like triangles, trapezoids, Pifunctions,
and Gaussians, but nothing bi-modal for example. We’ll make this more precise shortly. So
how do we do arithmetic with fuzzy numbers? The story goes like this.
Let A be a fuzzy subset of X. For each ∈(0,1]
A . The crisp set
α
A is
α , define
α
= { x ∈X A(x) ≥ α}
called the α-cut of A. The set 1 A , i.e., the set of x such that A(x) = 1, is called the core of A. Of course,
0
A
A , then
0+
A is the set
is all of X. If we define the strong α-level set of A by
α+
= { x ∈X A(x) > α}
of all x such that A(x) > 0, known as the support of A. Now we can formalize in the simple case what we
mean by convex fuzzy subsets of the reals, R. They are those fuzzy subsets all of whose α-cuts are crisp
convex subsets of the real numbers (intervals for one dimension). See [xxxKlir] for a more general
definition of a convex fuzzy set along with the theorem connecting that definition to α-cuts.