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1.1 Fuzzy Logic 3
Fuzzy sets provide means to model the uncertainty associated with vague-
ness, imprecision, and lack of information regarding a problem or a plant, etc.
Consider the meaning of a “short person.” For an individual X, the short per-
son may be one whose height is below 4
25
. For other individual Y, the short
person may be one whose height is below or equal to 3
90
. This “short” is
called as a linguistic descriptor. The term “short” informs the same meaning
to the individuals X and Y, but it is found that they both do not provide a
unique definition. The term “short” would be conveyed effectively, only when
a computer compares the given height value with the preassigned value of
“short.” This variable “short” is called as linguistic variable, which represents
the imprecision existing in the system.
The uncertainty is found to arise from ignorance, from chance and random-
ness, due to lack of knowledge, from vagueness (unclear), like the fuzziness
existing in our natural language. Lotfi Zadeh proposed the set membership
idea to make suitable decisions when uncertainty occurs. Consider the “short”
example discussed previously. If we take “short” as a height equal to or less
than 4 feet, then 3
90
would easily become the member of the set “short”
and 4
25
will not be a member of the set “short.” The membership value
is “1” if it belongs to the set or “0” if it is not a member of the set. Thus
membership in a set is found to be binary i.e., the element is a member of a
set or not.
It can be indicated as,
χ
A
(x)=
1
0
,x∈ A
,x /∈ A
,
where χ
A
(x) is the membership of element x in set A and A is the entire set
on the universe.
This membership was extended to possess various “degree of membership”
on the real continuous interval [0,1]. Zadeh formed fuzzy sets as the sets on the
universe X which can accommodate “degrees of membership.” The concept
of a fuzzy set contrasts with a classical concept of a bivalent set (crisp set),
whose boundary is required to be precise, i.e., a crisp set is a collection of
things for which it is known whether any given thing is inside it or not. Zadeh
generalized the idea of a crisp set by extending a valuation set {1, 0} (defi-
nitely in/definitely out) to the interval of real values (degrees of membership)
between 1 and 0 denoted as [0,1]. We can say that the degree of membership
of any particular element of a fuzzy set express the degree of compatibility of
the element with a concept represented by fuzzy set. It means that a fuzzy set
A contains an object x to degree a(x), i.e., a(x)=Degree(x ∈ A), and the
map a : X →{M embership Degrees} is called a set function or membership
function. The fuzzy set A canbeexpressedasA = {(x, a(x))}, x ∈ X,andit
imposes an elastic constrain of the possible values of elements x ∈ X called the
possibility distribution. Fuzzy sets tend to capture vagueness exclusively via
membership functions that are mappings from a given universe of discourse