Title: EMU, COMPUTER ENGINEERING DEPARTMENT 1999/2000 ACADEMIC YEAR, SPRING SEMESTER
1EMU, COMPUTER ENGINEERING DEPARTMENT1999/2000
ACADEMIC YEAR, SPRING SEMESTER
 C M P E 586
 Software Implementation of Fuzzy Systems
 PREPARED BY DR. KONSTANTIN DEGTIAREV
 FEBRUARY/JUNE 2000
 Slides use the material of books and journal
papers
2CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 1
Software Implementation of Fuzzy Systems
 ? ? ? Reference
 G.J.Klir, U.H.St.Clair, Bo Yuan. Fuzzy Set
Theory. Foundations Applications, Prentice Hall
PTR, 1997  B.Kosko. Fuzzy Engineering, Prentice Hall, 1997
 T.J.Ross. Fuzzy Logic with Engineering
Applications, McGrawHill, 1995  J.Yen, R.Langari. Fuzzy Logic. Intelligence,
Control, and Information, Prentice Hall, 1999  L.X.Wang. A Course in Fuzzy Systems and Control,
Prentice Hall, 1997  W.Pedrycz (ed.). Fuzzy Modelling. Paradigms and
Practice (Int. Series in Intelligent
Technologies), Kluwer Academic Publ., 1996  J.Yen. Fuzzy Logic  A Modern Perspective // IEEE
Transactions on Knowledge and Data Engineering,
vol.11, 1, January/February 1999  L.A.Zadeh. The Birth and Evolution of Fuzzy Logic
// Int. Journal on General
Systems, vol.17, 1990, pp.95105
1
3CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 2
Software Implementation of Fuzzy Systems
 ? ? ? Section1 Outline
 Introduction. Uncertainty, Imprecission and
Vagueness  Fuzzy Systems. Brief History of Fuzzy Logic.
Foundation of Fuzzy Theory.  Fuzzy Sets and Systems. Fuzzy Systems in
Commercial Products  Research fields in Fuzzy Theory
 the discussion of these topics takes
approximately 4 lecture hours. One example is
explained (CubiCalc? and fuzzyTECH? software
packages are used)
2
4CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 3
Software Implementation of Fuzzy Systems
 Most of the phenomena we encounter everyday are
imprecise  the imprecision may be associated
with their shapes, position, color, texture,
semantics that describe what they are  Fuzziness primarily describes uncertainty
(partial truth) and imprecision  The key idea of fuzziness comes from the
multivalued logic Everything is a matter of
degree  Imprecision raises in several faces, e.g. as a
semantic ambiguity
3
5CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 4
Software Implementation of Fuzzy Systems
 By fuzzifying crisp data obtained from
measurements, FL enhances the robustness of a
system  Imprecision raises in several faces  for
example, as a semantic ambiguity  the statement the soup is HOT is ambiguous,
but not fuzzy  e.g. 20º,80º
Definition of the domain of discourse
Transaction to Fuzziness
4
6CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 5
Software Implementation of Fuzzy Systems
 The word fuzzy can be defined as imprecisely
defined, confused, vague  Humans represent and manage natural language
terms (data) which are vague. Almost all answers
to questions raised in everyday life are within
some proximity of the absolute truth
5
7CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 6
Software Implementation of Fuzzy Systems
 Probability theory is one of the most traditional
theories for representing uncertainty in
mathematical models  Nature of uncertainty in a problem is a point
which should be clearly recognized by engineer 
there is uncertainty that arises from chance,
from imprecision, from a lack of knowledge, from
vagueness, from randomness  probability theory deals with the expectation of
an event (future event, its outcome is not known
yet), i.e. it is a theory of random events
6
8CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 7
Software Implementation of Fuzzy Systems
 Fuzziness deals with the impression of meaning of
concepts expressed in natural language  it is
not concerned with events at all  Fuzzy theory handles nonrandom uncertainty
7
9CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 8
Software Implementation of Fuzzy Systems
 As it is stated by L.Zadeh, in many cases there
is more to be gained from cooperation than from
arguments over which methodology is best  Many situations cover both kinds of uncertainty
 assume the weather forecast  tomorrow slight
rains are highly probable
8
10CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 9
Software Implementation of Fuzzy Systems
 The principle of incompatibility (L.Zadeh, 1973)
 As the complexity of a system increases, our
ability to make precise and yet significant
statements about its behavior diminishes until a
threshold is reached beyond which precision and
significance (or relevance) become almost
mutually exclusive characteristics
9
11CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 10
Software Implementation of Fuzzy Systems
 Intimate connection between fuzziness and
complexity (L.A.Zadeh)  a new approach to system analysis approximate
and yet effective means of describing the
behavior of systems which are too complex or too
illdefined to admit of precise mathematical
analysis
10
12CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 11
Software Implementation of Fuzzy Systems
 A new approach to system analysis a departure
from the conventional quantitative techniques of
system analysis  A new paradigm to develop approximate solutions
that are both costeffective and highly useful  a Fuzzy System (FS) is defined as a system with
operating principles based on fuzzy information
processing and decision making  There are several ways to represent knowledge,
but the most commonly used has a form of rules  IF (premise)A THEN (conclusion)B
11
13CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 12
Software Implementation of Fuzzy Systems
 From a knowledge representation viewpoint, a
fuzzy IFTHEN rule is a scheme for capturing
knowledge that involves imprecision  if we know
a premise (fact), then we can infer another fact
(conclusion)  A fuzzy system (FS) is constructed from a
collection of fuzzy IFTHEN rules  Acquisition of knowledge captured in IFTHEN
rules is NOT a trivial task (expert knowledge,
systems measurements, etc.)  The building blocks for fuzzy IFTHEN
rules are FUZZY SETS
12
14CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 13
Software Implementation of Fuzzy Systems
 The rule
 IF the air is cool THEN set the motor speed to
slow  has a form
 IF x is A THEN y is B,
 where fuzzy sets cool and slow are labeled
by A and B, correspondingly  A and B characterize fuzzy propositions about
variables x and y  Most of the information involved in human
communication uses natural language terms that
are often vague, imprecise, ambiguous by their
nature, and fuzzy sets can serve as the
mathematical foundation of natural language
13
15CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 14
Software Implementation of Fuzzy Systems
 A Fuzzy Set is a set with a smooth boundaries
 Fuzzy Set Theory generalizes classical set
theory to allow partial membership  Fuzzy Set A is a universal set U is determined by
a membership function ?A(x) that assigns to each
element x?U a number A(x) in the unit interval
0,1  Universal set U (Universe of Discourse) contains
all possible elements of concern for a particular
application  Fuzzy set has a onetoone correspondence
with its membership function
14
16CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 15
Software Implementation of Fuzzy Systems
 Fuzzy set A is defined as
 A (x, A(x)) , x?U, A(x)?0,1
 A(x) Degree(x?A) is a grade of membership of
element x?U in set A
15
17CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 16
Software Implementation of Fuzzy Systems
 The membership functions themselves are NOT fuzzy
 they are precise mathematical functions once a
fuzzy property is represented by a membership
function, nothing is fuzzy anymore  Suppose U is the interval 0,85 representing the
age of ordinary human beings, and the linguistic
term young as a function of age (value of the
variable age) can be defined as  see the graphical representation on the next
slide  !! pay attention to the usage of the symbol
/
16
18CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 17
Software Implementation of Fuzzy Systems
 If U is a set of integers from 1 to 10 (
U1,2,,10 ), then small is a fuzzy subset of
U, and it can be defined using enumeration
(summation notation)  A small 1/11/20.85/30.75/40.5/50.3/60.
1/7
Universe of discourse U is continuos
17
19CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 18
Software Implementation of Fuzzy Systems
 In the previous example elements of U (universal
set) with zero membership degrees are not
included into enumeration  A notion of a fuzzy set provides a convenient way
of defining abstraction  a process which plays a
basic role in human thinking and communication  All theories that use the basic concept of fuzzy
set can be called in a whole Fuzzy Theory  Rough classification of Fuzzy Theory can be
depicted as follows note that dependencies
between the branches are not shown
18
20CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 19
Software Implementation of Fuzzy Systems
19
21CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 20
Software Implementation of Fuzzy Systems
 The idea of Fuzzy Sets appeared in 1964
L.A.Zadeh (Professor of the University of
California at Berkeley) We need a radically
different kind of mathematics, the mathematics of
fuzzy or cloudy quantities which are not
described in terms of probability distributions  The paper Fuzzy Sets (Zadeh L.A., Information
and Control, vol.8, pp.338353, 1965) first used
the word fuzzy to mean vague in technical
literature  criticized by academic community the idea caused
a development of fuzzy set theory foundation
(19651980)  academic research work stimulates first
industrial applications of fuzzy systems
(19771990)  cement kiln controller (Denmark),
train control system (Sendai subway, Japan),
digital and analog fuzzy chips (USA, Japan)
20
22CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 21
Software Implementation of Fuzzy Systems
 Currently, the application fields of fuzzy
systems cover signal processing, communications,
expert systems, medicine, business/finance,
control (industrial processes and consumer
electronics),  widening of collaboration between universities
and industry, fuzzy boom (1987present) Japan
? Europe ? USA  1992 1st IEEE International Conference on Fuzzy
Systems  appearance of software companies (INFORM,
Aptronix,etc.)  Fuzzy Logic Toolbox for MATLAB was released in
1994  Courses on fuzzy sets and systems in Universities
curricula  Engineering consists largely of recommending
decisions based on insufficient information....
It is essential that these students be exposed to
ways of treating uncertainty and vagueness. This
also requires that existing faculty utilize these
methods  (Colin Brown, conference of NAFIPS)
21
23CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 22
Software Implementation of Fuzzy Systems
 Appearance of the new computational paradigms and
intensification of research in certain areas
(genetic algorithms/evolutionary strategies,
neural networks)  L.A.Zadeh introduced a term soft computing (1992)
  EXAMPLE 1 
 Fuzzy Toolbox Demo (MATLAB)
 by Dr.R.Babuška (Delft University of Technology,
The Netherlands)  IfThen Rules. Fuzzy reasoning (example)
 Word 97 document (preliminary explanations)
 end of the Section 1
22
24CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 23
Software Implementation of Fuzzy Systems
 ? ? ? Section 2 Outline
 Mathematical Background of Fuzzy Systems.
Classical (crisp) vs. Fuzzy Sets. Representation
of Fuzzy Sets  Types of Membership Functions. Basic concepts
(support, singleton, height, ?cut, convexity).
Fuzzy Set Operations  S and Tnorms. Properties of Fuzzy Sets. Sets as
points in Hypercubes. Cartesian Product. Crisp
and Fuzzy Relations  Linguistic variables and hedges. Membership
function design (shape analysis)  the discussion of these topics takes
approximately 10 lecture hours. Examples are
explained using CubiCalc, fuzzyTECH and FL
Toolbox packages
23
25CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 24
Software Implementation of Fuzzy Systems
 Fuzzy Systems F ?n ? ?p use m rules to map
vector input x to vector or scalar outputs F(x)  Fuzzy (Rulebased) Systems make use of linguistic
variables in their antecedents and consequents  Linguistic variables can be naturally represented
by fuzzy sets and logical connectives of these
sets
input X
24
26CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 25
Software Implementation of Fuzzy Systems
 A classical (crisp) set A in the universe of
discourse U can be defined in three ways   by enumerating (listing) elements (often
called list or extensional definition)   by specifying the common properties of
elements (intensional or rule definition)  the notation A x P(x) means that set A is
composed of elements x such that every x has the
property P(x)   by introducing a zeroone membership function
(characteristic or indicator definition)
25
27CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 26
Software Implementation of Fuzzy Systems
 Crisp set is a set with precise boundary, and
classical set theory is founded on the idea that
we can make crisp, exact distinctions between two
groups, i.e. between those individuals (elements)
that are definitely in the result set (group 1),
and those that are definitely outside it (group
2)  The basic operations on classical sets (A and B
are crisp sets in the universe of discourse U)  complement divides the universal set U into
 2 (two) parts
26
28CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 27
Software Implementation of Fuzzy Systems
 Fundamental properties of the basic operations
(these properties are also encountered in
propositional logic)
27
29CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 28
Software Implementation of Fuzzy Systems
 Fundamental properties satisfy to a principle of
duality replacing of empty set, U, ?, ? with
U, empty set, ?, ?, respectively, brings again
valid property  The notion of membership in fuzzy sets becomes a
matter of degree (number in the closed interval
0,1)  Membership of an element from the universe in
fuzzy set is measured by a function that attempts
to describe vagueness and ambiguity
28
30CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 29
Software Implementation of Fuzzy Systems
 Membership functions can be represented (a)
graphically, (b) in a tabular or list form, (c )
analytically and (d) geometrically (as a points
in the unit cube)  Geometrical representation for twoelement
universal set U (x1,x2) has a following
vizualization
29
31CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 30
Software Implementation of Fuzzy Systems
 see the previous figure Vertices (0,0),
(0,1), (1,0) and (1,1) represent all crisp sets
that can be defined for the universal set U, e.g.
the point (1,0) corresponds to the crisp set x1
(element x2 has no membership)  Membership functions can be symmetrical or
asymmetrical, and the most commonly used forms
are triangular, trapezoidal, Gaussian and bell
(the first two dominate in applications due to
simplicity and computational efficiency)  Membership functions are typically defined on
onedimensional universes, and in most cases, the
membership function appears in the continuos form  Fuzzy Toolbox Demo (MATLAB)
 by Dr.R.Babuška (Delft University of Technology)
 FuzzyTECH and CubiCalc (explanations)
30
32CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 31
Software Implementation of Fuzzy Systems
 The height of a fuzzy set A is the highest
(maximum) value of its membership function, i.e.
height(A)  If a fuzzy set has a height 1, then it is called
a normal fuzzy set in contrast, if height(A) lt
1, the fuzzy set is said to be subnormal  A subnormal set is a fuzzy set that contains only
elements with partial (lt1) membership  In most of applications fuzzy sets are normal,
and during the reasoning process usually
subnormal fuzzy sets are generated  A set of all elements of the universal set U
whose degree of membership in a fuzzy set A is
nonzero is called the support of a fuzzy set A,
i.e. supp(A)
31
33CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 32
Software Implementation of Fuzzy Systems
 A set of all elements x of the universal set U
with a property ?A(x) 1 (A is a fuzzy set) is
called the core of a fuzzy set A (core(A))
32
34CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 33
Software Implementation of Fuzzy Systems
 A fuzzy set whose support is a single point in
the universe of discourse U is called a fuzzy
singleton  Each fuzzy set A is associated with a family of
crisp subsets of A  their elements have such
membership degrees that they are restricted to a
crisp subset of 0,1  A crisp set A? that contains those x?U for which
is called an ?cut of a fuzzy
set A  The general property of ?cuts for any fuzzy
set A and two values ?1, ?2 ?0,1 that satisfy
to the condition ?1lt ?2 the following is true
33
35CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 34
Software Implementation of Fuzzy Systems
 Fuzzy sets may be completely characterized by
their ?cuts 
 (decomposition
theorem of fuzzy sets)  Example (Lecture
hours)
34
36CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 35
Software Implementation of Fuzzy Systems
 Consider a fuzzy set A which is represented
analytically in the universe of discourse U
5,15 as follows
Example
35
37CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 36
Software Implementation of Fuzzy Systems
 step 1 Several particular values of ? are
chosen from the unit interval 0,1  they are
0.1, 0.3, 0.5, 0.7 and 0.9  step 2 converting each of the ?cuts A? to
fuzzy sets for each x?U using the formula  (fuz_set?) ??A?(x)
Sometimes the theorem is referred as resolution
principle (approximate representation of
membership function)
Example
36
38CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 37
Software Implementation of Fuzzy Systems
 Fuzzy set A is convex if for any elements x1, x2
and x3 from the universal set U, the relation x1lt
x2lt x3 implies that  General property the intersection of two convex
sets produces a convex set  ? Convexity and ?cuts

37
39CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 38
Software Implementation of Fuzzy Systems
 Generalization of set operations to fuzzy sets is
not obvious  Operations on fuzzy sets are crucial to the fuzzy
inference process  In the rule IF (A or B) THEN C the true value
of C is the true value of the disjunction
(operation or)  Assume two fuzzy sets A and B are defined on the
universe of discourse U  three basic operations
can be represented as follows  Fuzzy set A is equal to fuzzy set B if and only
if  ?A(x) ?B(x), ?x?U
38
40CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 39
Software Implementation of Fuzzy Systems
 Fuzzy sets overlap with their complements (an
element  may partially belong to both fuzzy set and
sets complement)  In contrast, classical (crisp) sets never
overlap with their  complements
39
41CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 40
Software Implementation of Fuzzy Systems
 Two fundamental laws of Classical Set theory 
law of Excluded Middle and
law of Contradiction  ? are violated in Fuzzy Set
Theory (!!)  Standard fuzzy operations are quite adequate in
many practical applications of FS, but they do
not utilize the real expressive power of fuzzy
sets (what are the other possibilities that may
satisfy the requirements of practice? )  In practice, algebraic sum (1) and algebraic
product (2) are used for a definition of union
and intersection of two fuzzy sets, respectively
40
42CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 41
Software Implementation of Fuzzy Systems
 General notation
 Operator s is called an snorm if it satisfies to
the following axioms for any x, y, z and w
?0,1  Some of the operators (snorms) that model
(i.e. extend) fuzzy union
see the next slide...
41
43CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 42
Software Implementation of Fuzzy Systems
 (S1) Drastic sum
 (S2) Hamacher sum
 (S3) DuboisPrade class
 (S4) Yager class
 Note for arbitrary fuzzy sets A and B
membership values x and y stand for ?A(x) and
?B(x), correspondingly
Continuation tnorms (triangular norms)
42
44CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 43
Software Implementation of Fuzzy Systems
 Operator t is called an tnorm (triangular norm)
if it satisfies to the following axioms for any
x, y, z and w ?0,1  Some of the operators (tnorms) that model
(extend) fuzzy intersection  (T1) Drastic product
 (T2) Hamacher product
More still to come...
43
45CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 44
Software Implementation of Fuzzy Systems
 (T3) DuboisPrade class
 (T4) Yager class
 Selfstudying exercise Prove that the Yager
tnorm (class T4) converges to the min operator
when the parameter ? is in the infinite limit 
 An important properties of snorms and tnorms
can be summarized as follows
44
46CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 45
Software Implementation of Fuzzy Systems
 snorms are bounded below by max (standard fuzzy
union) and bounded above by drastic sum (S1)  tnorms are bounded below by drastic product (T1)
and bounded above by min (standard fuzzy
intersection)  snorms (a set of fuzzy disjunction operators)
are often called triangular conorms or shortly,
tconorms  The alternative forms of operators AND and OR are
called compensatory operators (they compensate
the  strictness of min and max operators proposed
by  L.A.Zadeh)
?
?
45
47CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 46
Software Implementation of Fuzzy Systems
 Operator c is called a fuzzy complement if it
satisfies to the following axioms for any x and y
?0,1  Some of the operators that model (extend) fuzzy
complement  (C1) Sugenos complement
 (C2) Yagers complement
 Demonstration (MATLAB environment)
46
48CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 47
Software Implementation of Fuzzy Systems
 Main types of membership functions (MF)
 (a) Triangular MF is specified by 3 parameters
a,b,c  (b) Trapezoidal MF is specified by 4 parameters
a,b,c,d
More to come...
47
49CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 48
Software Implementation of Fuzzy Systems
 (c) Gaussian MF is specified by 2 parameters
a,?  (d) Bellshaped MF is specified by 3 parameters
a,b,?  (e) Sigmoidal MF is specified by 2 parameters
a,b
48
50CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 49
Software Implementation of Fuzzy Systems
Image form NeuroFuzzy and Soft Computing
(J.S.R.Jang, C.T.Sun, E.Mizutanani 
supplementary slides
49
51CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 50
Software Implementation of Fuzzy Systems
 Assume X and Y are two arbitrary classical sets.
The Cartesian product of sets X and Y is a set of
all ordered pairs (xi,yj), xi?X, yj?Y that is  Suppose X x1, x2, x3, x4 , Y y1, y2, y3
the set XxY consists of 12 ordered pairs
(xi,yj), i1,2,3,4, j1,2,3  in this case, a
graphical representation (as nodes of a grid) is
convenient  Generalization of Cartesian product to n
arbitrary classical sets X1, X2,, Xn
50
52CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 51
Software Implementation of Fuzzy Systems
If the cardinality of the set X is n(X) and the
cardinality of the set Y is n(Y), then the
cardinality of the Cartesian product (set of
elements) is n(XxY) n(X)n(Y)
See the previous slide...
51
53CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 52
Software Implementation of Fuzzy Systems
 Omitting superscripts, an ordered sequence of n
elements (x1, x2, , xn) is called an ordered
ntuple  A subset of the Cartesian product
is called an nary relation built
over domains (sets) X1,X2,, Xn  if n2, the
binary relation on X1 and X2 (from X1 to X2) can
be formally defined as a set of ordered pairs in
X1xX2 that is  where P(x1,x2) is a property to which each pair
(x1,x2)?? satisfies  Example Suppose that both X1 and X2 are sets
of real  numbers ?5,20, i.e. X1
X2 ?5,20. The binary relation ?
? ?(X1,X2) less than has
52
54CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 53
Software Implementation of Fuzzy Systems
 the following formal analytical representation
 Graphical form of the binary relation ?
 A binary relation ? can be also represented by
53
55CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 54
Software Implementation of Fuzzy Systems
 means of membership function
 (arbitrary nary relation is a mapping
 ?(X1,X2,, Xn) X1xX2x xXn ? 0,1 )
 If a set X1xX2 is finite, then the values of
function ?? can be collected into a relational
matrix  Relations are intimately involved in logic,
approximate reasoning, rulebased systems, etc.  A rule IF x is A THEN y is B describes a
relation between the variables x and y  as
implication A ? B, rule expresses a
mapping (subset of Cartesian product)
between input and output domains
54
56CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 55
Software Implementation of Fuzzy Systems
 A fuzzy relation generalizes the concept of
classical (crisp) relation introducing a degree
of membership for each ordered ntuple (x1,
x2,, xn) in  For 2D case it can be defined as follows
 Examples
 a) x1 is close to x2 (both x1 and x2 are
numbers)  b) if x1 is medium, then x2 is high (x1 is an
observed state,  whereas x2 is a result state or action)
 c) x1 is similar to x2 (x1 and x2 can be
objects, human beings, properties) 
55
57CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 56
Software Implementation of Fuzzy Systems
 Formally, fuzzy relation in
can be defined as a fuzzy set  where is a mapping
 In different sources notations and can
be used for crisp and fuzzy relations,
correspondingly (if it is clear from the contents
which of two relations is used, sign can be
dropped)  means membership
degree of the ordered ntuple in the fuzzy
relation , where  , or a degree to
which fuzzy relation holds true for objects
 Example
(Lecture hours) 
56
58CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 57
Software Implementation of Fuzzy Systems
 Suppose that X1 and X2 are line segments 0,50
and 20,40, respectively, on the set of real
numbers . A fuzzy relation
x1 is approximately equal to x2 may be
defined by the membership function
Example
Fuzzy relations enhance our capability to deal
with relational concepts expressed in a natural
language !
57
59CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 58
Software Implementation of Fuzzy Systems
 A fuzzy relation x1 is much larger than x2 may
be defined by the membership function
Membership values line segment 0,1
Fuzzy relations are also fuzzy sets, and
fundamental properties of fuzzy sets hold for
fuzzy relations as well
58
60CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 59
Software Implementation of Fuzzy Systems
 Fuzzy set operations (complements, unions,
intersections) are applicable to fuzzy relations
too
Sugenos complement (parameter ? 2)
Standard complement
59
61CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 60
Software Implementation of Fuzzy Systems
 Operations on fuzzy sets defined on different
universal domains produce a multidimensional
fuzzy set (the following shows graphically the
result of operation )
60
62CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 61
Software Implementation of Fuzzy Systems
IF x is Ai ...
Example (Lecture hours)
THEN y is Bi
Contour plot
(MATLAB 5.2 environment)
Fuzzy sets Ai and Bi have Gaussian and
bellshaped forms
61
63CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 62
Software Implementation of Fuzzy Systems
 A fuzzy rule is described formally by a fuzzy
relation between antecedent and consequent, and
it defines a partition (fuzzy relation ) in
the space (Cartesian product). The union
of all partitions  forms a fuzzy graph (term introduced by
L.A.Zadeh) of a fuzzy model. The more partitions
(patches) the more accurate description of the
functional dependency Yf(X), where f is a crisp
function  Fuzzy system approximates a crisp function f by
means of patches the approximation is uniform,
and it allows specification of error level
(accuracy) in advance
62
64CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 63
Software Implementation of Fuzzy Systems
 A patch covering leads to major serious problem
of fuzzy systems exponential explosion of rules
(number of rules) as a result of increasing the
number of input variables  Fuzzy Approximation Theorem (B.Kosko, L.A.Zadeh)
A function can be approximated to any prescribed
accuracy provided that sufficient fuzzy rules
are available S.Wu and M.J.Er
References 1. L.Wang. Fuzzy Systems are
Universal Approximators // Proc.
Int.Conf.Fuzzy Syst., 1992
2. B.Kosko. Fuzzy Systems as Universal
Approximators // IEEE Transactions on
Computers, vol.43, 11, 1994
63
65CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 64
Software Implementation of Fuzzy Systems
 In a framework of fuzzy rules as conjunctions (IF
x is Ai THEN y is Bi ? Ai ? Bi ? Ai ? Bi  (functional relationship between input and
output on a base of linguistic terms, i.e. Ai and
Bi, i 1,n)  (A) Mamdani implication
 (correlationminimum)
 (B) Larsen implication
 (correlationproduct dilution of
membership values that are both small) 
cases (A) and (B)  Compound fuzzy propositions (e.g. x1 is Ai1 and
 x2 is Ai2) are interpreted as fuzzy
relations
Demonstration (MATLAB environment)
64
66CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 65
Software Implementation of Fuzzy Systems
 (1) x1 is Ai1 and x2 is Ai2
,  where t 0,1?0,1 ? 0,1 is a tnorm
 (2) x1 is Ai1 or x2 is Ai2
,  where s 0,1?0,1 ? 0,1 is a snorm
 As an implication, fuzzy rules represents human
abilities of imprecise reasoning  Fuzzy rule is an implication between fuzzy
propositions  IF ltfuzzy propositiongtAi THEN ltfuzzy
propositiongtBi (Ai ? Bi )  Fuzzy rule describes an entailment of Bi by Ai
(Ai entails Bi)  Fuzzy logic can be considered as a generalization
 of a classical binary and multivalued logic
65
67CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 66
Software Implementation of Fuzzy Systems
 The foundation of a fuzzy implication rule is
the narrow sense of fuzzy logic  In the classical propositional calculus (algebra
of propositions), implication P?Q, where P and Q
are two simple propositions, is logically
equivalent (gives the same truth table values) to  (1) and
 (2)
 As Ai and (or) Bi have fuzzy predicates,
 the implication becomes a fuzzy implication
 Twovalued logic
forms (1) and (2)
66
68CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 67
Software Implementation of Fuzzy Systems
 For the crisp propositions P and Q the
implication P?Q becomes global, i.e. the truth
table covers all possible combinations of 0/1
values of P and Q)  Fuzzy implication can be considered
as a local one, i.e. it gives a large truth value
only when both and have large truth
values  Having a rule IF x is Ai THEN y is Bi, we may
propose a following interpretation  IF x is Ai THEN y is Bi ELSE ltnothinggt
 (each rule covers a local part of the whole
working space)  Implication as a conjunction Ai ? Bi ? Ai ? Bi
(Mamdani (min) and Larsen (product) slide 64
implications are the most widely used in fuzzy
systems and fuzzy control)
67
69CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 68
Software Implementation of Fuzzy Systems
 Fuzzy rules can be considered as not being local
as well, and this fact opens a way to
classification of fuzzy rules as  (A) fuzzy mapping rules (FMR)
 (B) fuzzy implication rules (FIR)
 Case (A) FMR describe a functional dependency
between systems inputs and outputs by means of
linguistic terms (a totality of rules is
represented by a fuzzy graph). A collection of
fuzzy mapping rules are often called fuzzy model  Case (B) FIR describe an implication logic
relationship between two fuzzy propositions that
use linguistic terms and hedges (generalization
of twovalued logic)
similarities
 Both types of rules
 are represented as a fuzzy relations between
antecedent and consequent parts  use compositional rule of inference as an
inference engine
68
70CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 69
Software Implementation of Fuzzy Systems
differences

!  A major stages in fuzzy modeling are as follows
 1. fuzzy partition
 2. mapping of regions to local models
 3. fusing of local models into a global model
 4. defuzzification (quantitative summary of
possibility
distribution of models output)  The representatives of fuzzy implication families
(FI) are shown on the next slide (form FI1 is
also refered in  the literature as DienesRescher implication)
 But
 the semantics of relations is different
 different operators are used in compositional
rule of inference  two types of rules behave the same in the case
when antecedent parts of rules are satisfied, but
the behavior is different if antecedents are not
satisfied
Fuzzy RuleBased Modeling
69
71CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 70
Software Implementation of Fuzzy Systems
 Selected fuzzy implication (FI) operators
 (FI1) Zadehs classical maximum FI
 (FI1) Zadehs classical maximum FI
 (FI2) Lukasiewicz implication
 (FI3) Godelian FI
 (FI4) Standard sequence implication
 (it is shown
on the next slide)
equivalent to (FI1) when
70
72CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 71
Software Implementation of Fuzzy Systems
 Standard sequence implication
 Although some have been more extensively used,
there is no such thing as the fuzzy implication
operator and any practical application should
consider different alternatives, checking the
effectiveness of each one (A.Kandel, R.Pacheco,
A.Martins, S.Khator. The foundations of
rulebased computations in fuzzy models )  (fuzzy implication operators (1) Zadehs, (2)
Lukasiewicz, (3) Godelian)  PS. Godelian fuzzy implication is also referred
in the literature  as Brouwerian implication )
Demonstration (MATLAB environment)
71
73CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 72
Software Implementation of Fuzzy Systems
 The relationship between implications
 Two questions of interest
 (1) What are the criteria for choosing a
combination of fuzzy operators ?, ? and ? in
order to obtain a certain form of fuzzy
implication? For example, Lukasiewicz
implication (FI2) is a member of a family that
generalizes a material implication of a classical
logic (disjunction ?) Yagers snorm with ?
equal to 1 and (complement ?) standard
complementation 1  (2) For the fuzzy implication what
we mean  by equivalence to and
?
72
74CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 73
Software Implementation of Fuzzy Systems
 When a fuzzy relation is NOT finite, and it is
defined in ndimensional space, we have to
specify it analytically (through appropriate
formula)  A fuzzy relation on a finite Cartesian product
X1xX2 is usually represented by a fuzzy
relational matrix with elements taken from 0,1  Example Assume X1 a1, a2, a3 , X2 b1,
b2, b3, b4  are two sets of cities.
The relational concept close (distance
expressed in Km) is represented by the
(3,4)matrix
73
75CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 74
Software Implementation of Fuzzy Systems
 Two kinds of questions we can keep in mind
???  a) what is the degree that particular pair of
cities are considered to be close to each other?
(membership function of a fuzzy set)  b) what is a possibility that a short distance
(closeness to each other) corresponds to a
specific pair of city ai ( ) and city bj
( )? (possibility distribution of a
short distance (closeness))  Composition of relations
 Composition of two crisp binary relations R1 and
R2 requires their compatibility
Crisp case
74
76CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 75
Software Implementation of Fuzzy Systems
 Relation TR1R2 (composition of two crisp
relations) consists of those pairs (x,z) ,x?X,
z?Z, of the Cartesian product XxZ that via the
given relations R1 and R2 share at least one
element y?Y  Example Assume that X 20,40 , Y 0,50
,  Z 10,40 X,Y,Z ? ?
(set of real numbers). Two crisp relations R1 ?
XxY and R2 ? YxZ are defined (shown on the next
slide), and it is required to find their
composition 
 relation T  (as a result of compositional operation) is a
subset  of the Cartesian product XxZ
(see the next slide for the graphical
representation)
75
77CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 76
Software Implementation of Fuzzy Systems
Cartesian product XxY
line zy
relation R2 ? YxZ
Both relations are defined in twodimensional
Euclidean space
76
78CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 77
Software Implementation of Fuzzy Systems
 Two common forms of composition operation
 a) maxmin composition
 or in terms of 0/1 membership functions
 b) maxproduct (in general, maxstar)
composition  For example, maxproduct composition has a
following form
sign ? denotes any tnorm
77
79CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 78
Software Implementation of Fuzzy Systems
 It is stressed that maxmin method of
composition effectively expresses the approximate
and interpolative reasoning used by humans when
they employ linguistic propositions for deductive
reasoning (T.Ross. Fuzzy Logic with Engineering
Applications, McGrawHill, 1995 N.Vadiee.
Fuzzy rule based expert systems I, 1993)  A main goal of fuzzy logic is to form a
foundation for reasoning (inference) with
imprecise propositions such reasoning is called
approximate reasoning  Consider a given rule IF x is A THEN y is B,
where A and B are fuzzy sets (fuzzy propositions,
fuzzy predicates), and a fact x is A (A and A
are not necessarily identical). The result
produced by fuzzy inference engine  y is B A?R, where R is a fuzzy relation
which  represents an implication (x is A) ? (y is B)
78
80CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 79
Software Implementation of Fuzzy Systems
 An ordinarylanguage term represented by a fuzzy
set is called linguistic value  A linguistic variable can be considered as a
composition of a symbolic variable and a numeric
variable  Linguistic variable is a fundamental element in
human knowledge representation  Transition from crisp mathematics to fuzzy
mathematics by means of fuzzy set theory has
allowed mathematical representations to become
compatible with expressions in natural language  Linguistic hedges are special linguistic terms by
which other (primary) linguistic terms are
modified  Linguistic hedge (modifier) may be interpreted as
an unary operator that modifies
the meaning of a fuzzy set
79
81CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 80
Software Implementation of Fuzzy Systems
 Some of the most commonly used operators (hedges)
and their functions are as follows
Be cautious when treating the meaning of NOT
Functions of hedges
80
82CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 81
Software Implementation of Fuzzy Systems
 Brief comments on the previously mentioned
functions  (A) Concentration function keeping the
original shape (form) of a membership function,
shrink it over the universe of discourse, and a
level of concentration can be adjusted by
changing a value of power (gt1) applied to
membership values)  (B) Dilution function opposite to
concentration function it results in
spreading of membership function over universal
set through changing a power (lt1) of MF values  (C) Contrast intensification changes slightly
a shape of membership function, widening MF for
possibility values gt 0.5 and narrowing it when
the latter is ? 0.5  (D) Negation (not mentioned above) mirrors
imaging  of MF with respect to ?(x)0.5
81
83CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 82
Software Implementation of Fuzzy Systems
Negation
Contrast intensification
Concentration dilution
82
84CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 83
Software Implementation of Fuzzy Systems
 A definition of linguistic variable proposed by
L.A.Zadeh (The concept of a linguistic variable
and its application to approximate reasoning
I,II, Information Sciences,8,pp.199251,301357)
can be formulated as follows  A linguistic variable is characterized by
 (a) its name N
 (b) a set L of linguistic values it can
take  (c) universal set U (physical domain)
in which it is defined  (d) a rule R that associates each
linguistic value of L with a fuzzy  set in U
 Some observations on MF shape analysis
 (1) The location and granularity (number) of
MFs are the two relatively more important (from
the standpoint  of affecting performance of the fuzzy inference
83
85CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 84
Software Implementation of Fuzzy Systems
 algorithm) issues compared to the shape of each
member ship function  (2) The shape of MF characterizes uncertainty in
the fuzzy variable in general, a high level of
detail in shape design is considered as a
conceptual error  (3) Most of applications nowadays use simple
convex membership functions (due to computational
simplicity and relative easiness of
implementation in hardware, the commonly
practical are piecewiselinear forms,
e.g. triangular and trapezoidal MFs)  (4) In most cases heights of membership
functions  of antecedent variables are equal to 1.0 (normal
 sets) if the heights of MFs in consequent part
of  the rules is less than 1.0, then it may cause
84
86CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 85
Software Implementation of Fuzzy Systems
 socalled paralysis of implication results

(5) Overlapping is an important design
consideration each antecedent MF should
overlap only with imme diate neighboring
membership functions, i.e.
85
87CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 86
Software Implementation of Fuzzy Systems
 formally it can be expressed as
. Overlap of MFs
determines a degree of cooperation (or, switching
degree) between corresponding rules  (6) Each consequent MF represents one rule a
height of MF determines the strength of
contribution from each rule, and MFs location
affects the actual decision value  (7) In general, shape modifications of
antecedent MF (compared to those of consequent
ones) produce more significant effects on the
output behavior  (8) Adjustment (symmetric changes in overlap) of
all consequent MFs do not produce significant
changes of outputs behavior  end of the Section 2
86
88CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 87
Software Implementation of Fuzzy Systems
 ? ? ? Section 3 Outline
 Basic Principles of Inference in Fuzzy Logic
(Entailment, Conjunction, Generalized Modus
Ponens, Generalized Modus Tollens). Fuzzy IFTHEN
rules. Canonical form  Fuzzy Systems and Algorithms. Approximate
Reasoning  Fuzzy Inference Engines. Graphical Techniques of
Inference. Fuzzification/Defuzzification  Fuzzy System Design and its Elements (conceptual
model). Design Options  the discussion of these topics takes
approximately 8?10 lecture hours. Examples are
explained using fuzzyTECH, FL Toolbox packages
and  MATLAB demonstrations
87
89CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 88
Software Implementation of Fuzzy Systems
 Contrariwise , continued Tweedledee, if it
was so, it might be and if it were so, it would
be but as it isnt , it aint. Thats logic
(Lewis Carroll, Through the Looking Glass)  In the propositional logic the inference can be
depicted as follows  If the resulting premises are both true, then the
conclusion is also true (the truth conclusion is
inferred or deduced from truth premises we call
it a valid deduction). For example, the following
form is invalid
 P and Q are propositions (variables)
 sign ? represents the relation ifthen
 symbol (therefore) is placed before
conclusion
premise 1 IF P THEN Q premise 2
not P
88
90CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 89
Software Implementation of Fuzzy Systems
 Among basic inference rules (the premises
logically imply rules conclusions) we can
mention the following  (a) Modus Ponens (b)
Modus Tollens  (Lat. ? method of affirming) (Lat. ?
method of denying)  (c) Hypothetical Syllogism (many mathematical
arguments contain a chain of ifthen statements)  A test for validity of Modus Tollens is as
follows
The implication connective (?) is especially
important as a basis of fuzzy implication rules
89
91CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 90
Software Implementation of Fuzzy Systems
 premises are conjuncted in the antecedent part
of implication, and conclusion form its
consequent part
How validity is checked?
Fuzzy Logic (FL) generalizes the notion of
truth values in classical logic, and provides a
background for reasoning (inferencing) when
90
92CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 91
Software Implementation of Fuzzy Systems
 corresponding conditions are only partially
satisfied  Approximate reasoning (original rule IF x is A
THEN y is B)  1. Possibility distribution of the variable x
 2. Implication possibility from x to y
 possibility
distribution of y  Both fuzzy implication and fuzzy mapping rules
use a compositional rule of inference for
calculation of output results, but there are
still some differences  If proposition P is described by set A ? X, and
proposition Q is described by set B ? Y, then the
classical implication P?Q can be represented by
the relation R as follows  its
schematical  representation (Venn diagram, Figure 1)
91
93CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 92
Software Implementation of Fuzzy Systems
 is as follows
 A rule IF x is A THEN y is B can be expressed
as a compound conditional statement IF x is A
THEN y is B ELSE y is Nothing (no action). In
general,  expression IF x is A THEN y is B ELSE y is C
See comments related to Figure 2 below
92
94CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 93
Software Implementation of Fuzzy Systems
 can be expressed as a disjunction
 IF x is A THEN y is B ? IF x is THEN
y is C  In the logic of compound statements this can be
written as 
, where S is a proposition  described by set C ? Y (see Venn diagram,
Figure 2)  Figure 1 (shaded area truth domain of the
implication P?Q)  Figure 2 (shaded area truth domain of the form
, where
)  Consider a given rule IF x is A THEN y is B,
where A and B are fuzzy sets (fuzzy propositions,
fuzzy predicates) defined on universes X and Y,
correspondingly,  and a fact x is A (A and A are not
necessarily  identical)
93
95CMPE 586
Prepared Dr.Konstantin Degtiarev, FebruaryJune
2000 Slide 94
Software Implementation of Fuzzy Systems
 Having a rule and a fact, what can we say about
conclusion (consequent) B ?  Generalized Modus Ponens
 Rule (premise 1) x is A ? y is B
 Fact (premise 2) x is A
 Infer (result produced by fuzzy inference
engine) y is B  B A ? R, where R is a fuzzy relation
(associations between the elements of