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

width provide a guide. The nurse turns the instrument over to recover what is effectively a membership
(after dividing by 10) in the fuzzy set “Pain”. The goal is to provide sufficient pain medication without
over dosing. This is a continuous membership. For very young children, see the discrete memberships as
shown in Figure 2.3(b).
Example x.x Definitions of membership functions:
It’s pretty obvious how to define triangular and trapezoidal fuzzy membership functions over a real
valued domain, i.e., an interval subset of R, (piecewise linear functions – right?). See problem X.xx. S-
functions are defined by 3 parameters (a,b,c) where a < b < c. The function S(x; a,b,c) is required to be 0
up to a; a parabola that opens up from a to b where S(b; a,b,c) = ½; a parabola that “matches up” and
opens downward from b to c with S(c; a,b,c) = 1. The equation for such an S-function is
⎧ 0
⎪ (x − a)
2
⎪
⎪ 2(b − a)
2
S(x;a,b,c) = ⎨
⎪−
(x − c)
2
+ 1
⎪
⎪
2(b − c)
2
⎩ 1
x ≤ a
a < x ≤ b
b < x ≤ c
x > c
Hmmm, how do we get that equation? Do problem X.xx. A Z-function is just the “flip” of the S-function,
i.e., Z(x;a,b,c) = 1 – S(x;a,b,c). A Pi function just pieces an S-function with a Z-function to look
something like a gaussian, though it actually reaches 0 and is made from parabolic sections. It has 6
parameters a < b < c ≤ d < e < f and is defined by
Pi(x; a,b,c,d,e,f)
⎧S(x;a,b,c)
⎪
= ⎨ 1
⎪
⎩Z(x;d,e,f )
x ≤ c
c < x < d
x ≥ d
Note that if c < d, a Pi function resembles a “soft” trapezoid. Of course, we can (and do) use properly
scaled Gaussian functions as membership functions in many applications. They have the advantage of
having derivatives of all orders, but they never actually reach 0.
Notice that all of these membership functions have at least one value where the function value is 1. A
fuzzy set A whose membership function A(x) “reaches” 1 is called normal. To be precise, define the
height of A by ht(A) = sup {A(x)}
. We use supremum (sup) instead of maximum to handle certain
x∈X
1
infinite domain cases, like the logistic membership function A(x) = 1 + e
−x
where the domain X is the