Fuzzy Logic and Fuzzy Inference
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Why use fuzzy logic?
Tipping example
Fuzzy set theory
Fuzzy inference
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What is fuzzy logic?
• A super set of Boolean logic
• Builds upon fuzzy set theory
• Graded truth. Truth values between True and False. Not everything
is either/or, true/false, black/white, on/off etc.
• Grades of membership. Class of tall men, class of far cities, class of
expensive things, etc.
• Lotfi Zadeh, UC/Berkely 1965. Introduced FL to model
uncertainty in natural language. Tall, far, nice, large, hot, …
• Reasoning using linguistic terms. Natural to express expert
knowledge.
If the weather is cold then wear warm clothing
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Fuzzy logic – A Definition
• Fuzzy logic provides a method to formalize reasoning when dealing
with vague terms.
• Traditional computing requires finite precision which is not always
possible in real world scenarios. Not every decision is either true or
false, or as with Boolean logic either 0 or 1.
• Fuzzy logic allows for membership functions, or degrees of
truthfulness and falsehoods. Or as with Boolean logic, not only 0
and 1 but all the numbers that fall in between.
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Why use fuzzy logic?
Pros:
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Conceptually easy to understand w/ “natural” maths
Tolerant of imprecise data
Universal approximation: can model arbitrary nonlinear functions
Intuitive
Based on linguistic terms
Convenient way to express expert and common sense knowledge
Cons:
• Not a cure-all
• Crisp/precise models can be more efficient and even convenient
• Other approaches might be formally verified to work
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Tipping example
• The Basic Tipping Problem: Given a number between 0 and 10
that represents the quality of service at a restaurant what should
the tip be?
Cultural footnote: An average tip for a meal in the U.S. is 15%,
which may vary depending on the quality of the service provided.
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Tipping example: The non-fuzzy approach
• Tip = 15% of total bill
• What about quality of service?
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Tipping example: The non-fuzzy approach
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Tip = linearly proportional to service from 5% to 25%
tip = 0.20/10*service+0.05
What about quality of the food?
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Tipping example: Extended
• The Extended Tipping Problem: Given a number between 0 and
10 that represents the quality of service and the quality of the food,
at a restaurant, what should the tip be?
How will this affect our tipping formula?
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Tipping example: The non-fuzzy approach
• Tip = 0.20/20*(service+food)+0.05
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We want service to be more important than food quality. E.g., 80% for
service and 20% for food.
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Tipping example: The non-fuzzy approach
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Tip = servRatio*(.2/10*(service)+.05) +
(1-servRatio)*(.2/10*(food)+0.05);
servRatio = 80%
Seems too linear. Want 15% tip in general and deviation only for
exceptionally good or bad service.
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Tipping example: The non-fuzzy approach
if service < 3,
tip(f+1,s+1) = servRatio*(.1/3*(s)+.05) + ...
(1-servRatio)*(.2/10*(f)+0.05);
elseif s < 7,
tip(f+1,s+1) = servRatio*(.15) + ...
(1-servRatio)*(.2/10*(f)+0.05);
else,
tip(f+1,s+1) = servRatio*(.1/3*(s-7)+.15) + ...
(1-servRatio)*(.2/10*(f)+0.05);
end;
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Tipping example: The non-fuzzy approach
Nice plot but
• ‘Complicated’ function
• Not easy to modify
• Not intuitive
• Many hard-coded parameters
• Not easy to understand
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Tipping problem: the fuzzy approach
What we want to express is:
1.
2.
3.
4.
5.
or
1.
2.
3.
If service is poor then tip is cheap
If service is good the tip is average
If service is excellent then tip is generous
If food is rancid then tip is cheap
If food is delicious then tip is generous
If service is poor or the food is rancid then tip is cheap
If service is good then tip is average
If service is excellent or food is delicious then tip is generous
We have just defined the rules for a fuzzy logic system.
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Tipping problem: fuzzy solution
Decision function generated using the 3 rules.
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Tipping problem: fuzzy solution
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Before we have a fuzzy solution we need to find out
a) how to define terms such as poor, delicious, cheap, generous etc.
b) how to combine terms using AND, OR and other connectives
c) how to combine all the rules into one final output
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Fuzzy sets
• Boolean/Crisp set A is a mapping for the elements of S to
the set {0, 1}, i.e., A: S  {0, 1}
• Characteristic function:
A(x) =
{
1
if x is an element of set A
0
if x is not an element of set A
• Fuzzy set F is a mapping for the elements of S to the interval
[0, 1], i.e., F: S  [0, 1]
• Characteristic function: 0  F(x)  1
• 1 means full membership, 0 means no membership and anything in
between, e.g., 0.5 is called graded membership
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Fuzzy Sets (contd.)
• fuzzy set A
• A = {(x, µA(x))| x Є X} where µA(x) is called the membership
function for the fuzzy set A. X is referred to as the universe of
discourse.
• The membership function associates each element x Є X with a
value in the interval [0,1].
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Example: Crisp set Tall
• Fuzzy sets and concepts are commonly used in natural language
John is tall
Dan is smart
Alex is happy
The class is hot
• E.g., the crisp set Tall can be defined as {x | height x > 1.8 meters}
But what about a person with a height = 1.79 meters?
What about 1.78 meters?
…
What about 1.52 meters?
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Example: Fuzzy set Tall
• In a fuzzy set a person with a height of 1.8 meters would be
considered tall to a high degree
A person with a height of 1.7 meters would be considered tall to a
lesser degree etc.
• The function can change
for basketball players,
Danes, women,
children etc.
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Membership functions: S-function
• The S-function can be used to define fuzzy sets
• S(x, a, b, c) =
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0
2(x-a/c-a)2
1 – 2(x-c/c-a)2
1
for x  a
for a  x  b
for b  x  c
for x  c
a
b
c
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Membership functions of one dimension
• These membership functions are some of the commonly used
membership functions in the fuzzy inference systems.
• Triangle(x; a, b, c) = 0 if x  a;
= (x-a)/(b-a) if a  x  b;
= (c-b)/(c-b) if b  x  c;
= 0 if c  x.
• Trapezoid(x; a, b, c, d) = 0 if x  a;
= (x-a)/(b-a) if a  x  b;
= 1 if b  x  c;
= (d-x)/(d-c) 0 if c  x  d;
= 0, if d  x.
• Sigmoid(x; a, c) = 1/(1 + exp[-a(x-c)]) where a controls slope at
the crossover point x = c.
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Membership functions of two dimensions
• One dimensional fuzzy set can be extended to form its cylindrical
extension on second dimension
• Fuzzy set A = “(x,y) is near (3,4)” is
• µA(x,y) = exp[- ((x-3)/2)2 -(y-4)2 ]
= exp[- ((x-3)/2)2 ] exp -(y-4)2 ]
=gaussian(x;3,2)gaussian(y;4,1)
• This is a composite MF since it can be decomposed into two
gaussian MFs
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Membership functions: P-Function
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P(x, a, b) =
• S(x, b-a, b-a/2, b)
• 1 – S(x, b, b+a/2, a+b)
for x  b
for x  b
E.g., close (to a)
a
b-a
b-a/2
a
b+a/2
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Simple membership functions
• Piecewise linear: triangular etc.
• Easier to represent and calculate  saves computation
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Other representations of fuzzy sets
• A finite set of elements:
F = 1/x1 + 2/x2 + … n/xn
+ means (Boolean) set union
• For example:
TALL = {0/1.0, 0/1.2, 0/1.4, 0.2/1.6, 0.8/1.7, 1.0/1.8}
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Fuzzy sets with a discrete universe
• Let X = {0, 1, 2, 3, 4, 5, 6} be a set of numbers of children a family
may possibly have.
• fuzzy set A with “sensible number of children in a family” may be
described by
• A = {(0, 0.1), (1, 0.3), (2, 0.7), (3, 1), (4, 0.7), (5, 0.3), (6, 0.1)}
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Fuzzy sets with a continuous universe
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X = R+ be the set of possible ages for human beings.
fuzzy set B = “about 50 years old” may be expressed as
B = {(x, µB(x)) | x Є X}, where
µB(x) = 1/(1 + ((x-50)/10)4)
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We use the following notation to describe fuzzy sets.
• A = Σ xi Є X µA(xi)/ xi, if X is a collection of discrete objects,
• A = ∫X µA(x)/ x, if X is a continuous space.
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Fuzzy set operators
• Equality
A=B
A (x) = B (x)
• Complement
A’
A’ (x) = 1 - A(x)
• Containment
AB
A (x)  B (x)
• Union
A B
A  B (x) = max(A (x), B (x))
• Intersection
AB
A  B (x) = min(A (x), B (x))
for all x  X
for all x  X
for all x  X
for all x  X
for all x  X
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• Support(A) is set of all points x in X such that
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{(x, µA(x)) | µA(x) > 0 }
• core(A) is set of all points x in X such that
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{(x, µA(x)) | µA(x) =1 }
• Fuzzy set whose support is a single point in X with µA(x) =1 is
called fuzzy singleton
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• Crossover point of a fuzzy set A is a point x in X such that
• {(x, µA(x)) | µA(x) = 0.5 }
• α-cut of a fuzzy set A is set of all points x in X such that
• {(x, µA(x)) | µA(x) ≥ α }
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Example fuzzy set operations
A’
A
AB
AB
A
B
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Linguistic Hedges
• Modifying the meaning of a fuzzy set using hedges such as very,
more or less, slightly, etc.
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Concentration or Con operator
Very F = F2
Dilation or Dil operator
More or less F = F1/2
tall
more or less tall
= DIL(old);
More or less tall
• extremely tall
Very tall
= CON(CON(CON(old)))
• etc.
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Fuzzy inference
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Fuzzy logical operations
Fuzzy rules
Fuzzification
Implication
Aggregation
Defuzzification
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Fuzzy logical operations
• AND, OR, NOT, etc.
• NOT A = A’
= 1 - A(x)
• A AND B = A  B = min(A (x), B (x))
• A OR B = A  B = max(A (x), B (x))
min(A,B)
From the following
truth tables it is
seen that fuzzy
logic is a superset
of Boolean logic.
max(A,B)
1-A
A
B
A and B
A
B
A or B
A
not A
0
0
0
0
0
0
0
1
0
1
0
0
1
1
1
0
1
0
0
1
0
1
1
1
1
1
1
1
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If-Then Rules
• Use fuzzy sets and fuzzy operators as the subjects and verbs of
fuzzy logic to form rules.
if x is A then y is B
where A and B are linguistic terms defined by fuzzy sets on the sets
X and Y respectively.
This reads
if x == A then y = B
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Evaluation of fuzzy rules
• In Boolean logic:
pq
if p is true then q is true
• In fuzzy logic:
pq
if p is true to some degree then q is true to some degree.
0.5p => 0.5q
(partial premise implies partially)
• How?
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Evaluation of fuzzy rules (cont’d)
• Apply implication function to the rule
• Most common way is to use min to “chop-off” the consequent
(prod can be used to scale the consequent)
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Summary: If-Then rules
1. Fuzzify inputs
Determine the degree of membership for all terms in the premise.
If there is one term then this is the degree of support for the
consequent.
2. Apply fuzzy operator
If there are multiple parts, apply logical operators to determine the
degree of support for the rule.
3. Apply implication method
Use degree of support for rule to shape output fuzzy set of the
consequent.
How do we then combine several rules?
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Multiple rules
• We aggregate the outputs into a single fuzzy set which combines their
decisions.
• The input to aggregation is the list of truncated fuzzy sets and the
output is a single fuzzy set for each variable.
• Aggregation rules: max, sum, etc.
• As long as it is commutative then the order of rule exec is irrelevant.
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max-min rule of composition
• Given N observations Ei over X and hypothesis Hi over Y we have N
rules:
if E1 then H1
if E2 then H2
if EN then HN
• H = max[min(E1), min(E2), … min(EN)]
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Defuzzify the output
• Take a fuzzy set and produce a single crisp number that represents
the set.
• Practical when making a decision, taking an action etc.
I=
 I x
 I
Center of gravity
Center of largest area
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Fuzzy inference overview
Tip = 16.7 %
Result of defuzzification
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Limitations of fuzzy logic
• How to determine the membership functions? Usually requires finetuning of parameters
• Defuzzification can produce undesired results
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Fuzzy tools and shells
• Matlab’s Fuzzy Toolbox
• FuzzyClips
• Etc.
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General Fuzzified Applications
• Quality Assurance
• Error Diagnostics
• Control Theory
• Pattern Recognition
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Specific Fuzzified Applications
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Otis Elevators
Vacuum Cleaners
Hair Dryers
Air Control in Soft Drink Production
Noise Detection on Compact Disks
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Cranes
Electric Razors
Camcorders
Television Sets
Showers
Japan, Korea, China were the early adapters of
Fuzzy Logic into industrial applications
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Expert Fuzzified Systems
• Medical Diagnosis
• Legal
• Stock Market Analysis
• Mineral Prospecting
• Weather Forecasting
• Economics
• Politics
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References
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Slides from CS 561 (Intro to AI) course, Sessions 20-21 of Paolo Pirjanian
Slides from OperMgt 345 course of Mitch Pence, Boise State University
Slides from CS623-Lec11-4Sept06, S.G.Sanjeevi of IIT Bombay.
Fuzzy Logic. Fuzzy Logic - a powerful new technology.
http://www.austinlinks.com/Fuzzy/
FuzzyNet On-line. Automatic Transmission
http://www.aptronix.com/fuzzynet/applnote/transmis.htm
Garner, Martin. Weird Water and Fuzzy Logic: More notes of a Fringe
Watcher.
Generation 5. An Introduction to Fuzzy Logic.
http://www.generation5.org/fuzzyintro.shtml
Sowell, Thomas . FUZzy Logic For “Just Plain Folks” .
You Fuzzin’ with me?
http://www.doc.ic.ac.uk/~nd/surprise_96/journal/vol1/sbaa/article1.html
Zadeh, Lotfi A.
http://http.cs.berkeley.edu/People/Faculty/Homepages/zadeh.html
Zadeh, L. (1965), "Fuzzy sets", Information and Control, 8: 338-353
Jang J.S.R., (1997): ANFIS architecture. In: Neuro-fuzzy and Soft
Computing (J.S. Jang, C.-T. Sun, E. Mizutani, Eds.), Prentice Hall, New
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Jersey
An Example
The following, illustrates a basic “fuzzy” automatic transmission
system. The transmission uses four fuzzy sensor inference
inputs to control the best gear selection for the given conditions.
The inputs are throttle, vehicle speed, engine speed and engine
load.
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An Example (cont.)
Labels and Membership Functions of Throttle.
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An Example (cont.)
Labels and Membership Functions of vehicle speed.
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An Example (cont.)
Labels and Membership Functions of engine speed.
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An Example (cont.)
Labels and Membership Functions of engine load.
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An Example (cont.)
Labels and Membership Functions of shift.
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An Example (cont.)
Using the labels as defined in the previous slides, rules can be
written for the fuzzy interface unit shown earlier. The rules provide
a tangible knowledge base required for the decision process and are
represented as English like if-then statements.
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An Example (cont.)
To create the fuzzy interface unit, rules such as the following would
be developed to facilitate the automatic shifting of the vehicle.
If vehicle speed is low, variation of vehicle speed is small, slope
resistance is positive large and accelerator is medium then mode is
steep uphill mode.
If vehicle speed is medium, variation of vehicle speed is small, slope
resistance is negative large and accelerator is small then mode is
gentle downhill mode.
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An Example (cont.)
If Mode is Steep uphill mode, the Shift is No. 2
If Mode is Gentle downhill mode, then Shift is No. 3
The previous slides illustrate how fuzzy logic can provide a powerful
tool when addressing complex situations that are not feasible using
conventional approaches. By employing fuzzy logic, we have the
ability to include additional variables and rules to take into account
factors that might improve the behavior of the control system.
* See reference (see notes):
http://www.aptronix.com/fuzzynet/applnote/transmis.htm
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Binary fuzzy relation
• A binary fuzzy relation is a fuzzy set in X × Y which maps each
element in X × Y to a membership value between 0 and 1. If X and
Y are two universes of discourse, then
• R = {((x,y), R(x, y)) | (x,y) Є X × Y } is a binary fuzzy relation in X
× Y.
• X × Y indicates cartesian product of X and Y
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Fuzzy Rules
• Fuzzy rules are useful for modeling human thinking, perception and
judgment.
• A fuzzy if-then rule is of the form “If x is A then y is B” where A
and B are linguistic values defined by fuzzy sets on universes of
discourse X and Y, respectively.
• “x is A” is called antecedent and “y is B” is called consequent.
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Fuzzy relations
• A fuzzy relation for N sets is defined as an extension of the crisp
relation to include the membership grade.
R = {R(x1, x2, … xN)/(x1, x2, … xN) | xi  X, i=1, … N}
which associates the membership grade, R , of each tuple.
• E.g.
Friend = {0.9/(Manos, Nacho), 0.1/(Manos, Dan),
0.8/(Alex, Mike), 0.3/(Alex, John)}
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Fuzzy intersection and Union
• AB = T(A(x), B(x)) where T is T-norm operator. There are some
possible T-Norm operators.
• Minimum: min(a,b)=a ٨ b
• Algebraic product: ab
• Bounded product: 0 ٧ (a+b-1)
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•
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C(x) = AB = S(A(x), B(x)) where S is called S-norm operator.
It is also called T-conorm
Some of the T-conorm operators
Maximum: S(a,b) = max(a,b)
Algebraic sum: a+b-ab
Bounded sum: = 1 ٨(a+b)
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Examples, for such a rule are
• If pressure is high, then volume is small.
• If the road is slippery, then driving is dangerous.
• If the fruit is ripe, then it is soft.
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• The fuzzy rule “If x is A then y is B” may be abbreviated as A→ B
and is interpreted as
A × B.
• A fuzzy if then rule may be defined (Mamdani) as a binary fuzzy
relation R on the product space X × Y.
• R = A→ B = A × B =∫X×Y A(x) T-norm B(y)/ (x,y).
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