The document discusses fuzzy implications, also known as fuzzy rules or fuzzy conditionals, which are essential for fuzzy control systems and describe the relationship between two linguistic values. It outlines various interpretations of fuzzy implications, different types of fuzzy relations, and essential operators used in their formulation. Key implications such as Kleene-Dienes, Lukasiewicz, Zadeh's, and Mamdani's are introduced, along with their mathematical definitions and significance in fuzzy logic.

1. EE-646
Lecture-9
Fuzzy Implications

2. Introduction
• A fuzzy rule generally assumes the form
R: IF x is A, THEN y is B. Where A and B are
linguistic values defined by fuzzy sets on UoD X
and Y, respectively.
• The rule is also called a “fuzzy implication” or
fuzzy conditional statement.
• Fuzzy implication is an important connective in
fuzzy control systems because the control
strategies are embodied by sets of IF-THEN rules
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3. Contd...
• Sometimes, the fuzzy rule is abbreviated as
R: A → B or simply A → B
• In essence, the expression describes a relation
between two variables x and y.
• This suggests that a fuzzy rule can be defined
as a binary relation R on the product space
X × Y
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4. Different Interpretations
1. “A is coupled with B”
is T-norm operator
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( ) ( )
( , )
A B
X Y
x y
R A B A B
x y
 


     


5. Interpretations
2. “A entails B”→ Four different formulae
a. Material Implication
b. Propositional calculus
c. Extended Propositional calculus
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R A B A B    
 R A B A A B     
 R A B A B B     

6. Interpretations
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d. Generalized Modus Ponens (GMP)
All the above four formulae reduce to the
familiar identity
 ( , ) sup | ( ) ( ),0 1R A Bx y c x c y c      
A B A B   

7. Contd...
• Based on these two interpretations & various
T-norm/co-norm operators, a no. of qualified
methods can be formulated to calculate the
fuzzy relation R: A → B
• Relation R can be viewed as fuzzy set with 2D
MF
• f is called fuzzy implication function &
performs the task of transforming the
membership grades of x in A & y in B into
those of (x, y) in A → B
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 ( , ) ( ), ( ) ( , )R A Bx y f x y f a b   

8. Contd...
• For the first interpretation, “A is coupled with
B” 4 different fuzzy relations A → B result from
employing the most commonly used T-norm
operators (min, algebraic, bounded, drastic
product)
• For the second interpretation, “A entails B”,
again 4 different fuzzy relations A → B have
been reported in literature (Zadeh’s arithmetic
rule, Zadeh’s max-min rule, Boolean,
Goguen’s)
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9. Different Implications
1. Kleene-Diene’s Implication
2. Lukasiewicz’s "
3. Zadeh’s "
4. Stochastic "
5. Goguen’s "
6. Gödel’s "
7. Sharp "
8. General "
9. Mamdani’s "
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10. 1. Kleene Diene’s Implication
Where, C is the cylindrical extension
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   K 'y yR C A C B 
 K
max 1 ( ), ( )R A Bx y   

11. 2. Lukasiewicz’s Implication
is bounded sum
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   L 'y yR C A C B 

   L
, min 1,1 ( ) ( )R A Bx y x y    

12. 3. Zadeh’s Implication
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     Z 'y y yR C A C B C A    
     Z
, max min ( ), ( ) , 1 ( )R A B Ax y x y x      

13. 4. Stochastic Implication
It is based on the following equality
Defined as
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 1 ( ) ( ) ( )
B
P P A P A P B
A
 
    
 
     St 'y y yR C A C A C B    
   st , min 1,1 ( ) ( ) ( )R A A Bx y x x y      

14. 5. Goguen’s Implication
One of the requirements in multi-valued logic is
that A → B should satisfy
This goal is achieved when we use the definition
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( ) ( , ) ( )A A B Bx x y y   
GN
GN min 1, ( ) ( )
( )
min 1,
( )
y y
A
R
B
R C A C B
x
y



   
 
  
 

15. 6. Gödel’s Implication
It is one of the best known implication formulae
in multi-valued logic. It is defined as:
e.g.
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1, ( ) ( )
( ),otherwiseg
A B
R
B
x y
y
 



 

(0.5,0.7) 1 and (0.8,0.6) 0.6
g g
A B A B   

16. 6. Gödel’s Implication...contd
Above definition results in the following fuzzy
relation that is frequently used in fuzzy logic
or,
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( ) ( )g y yg
R C A C B   
( , ) ( ) ( )gR A Bg
x y x y   

17. 7. Sharp Implication
Looks like Gödel’s but it is more restrictive. It is
defined as:
e.g.
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1, ( ) ( )
0, ( ) ( )s
A B
A B
A B
x y
x y
 

 

 

(0.5,0.7) 1 and (0.8,0.6) 0
s s
A B A B   

18. 7. Sharp Implication...contd
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( ) ( )s y ys
R C A C B   
( , ) ( ) ( )sR A Bs
x y x y   

19. 8. General Implication
A very general implication which does not have
any explicit name in the literature may be
considered as combination of Gödel and sharp.
It is based on the following formula:
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   A B A B A B 
     

20. 8. General Implication...contd
Where α and β may be s or g. In fuzzy terms, Rαβ
is defined as:
This relation is hardly ever used in the literature.
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 
    
( ) ( ) ,
( , ) min
1 ( ) 1 ( )
A B
R
A B
x y
x y
x y



 

 
 
 
  
   
 

21. 9. Mamdani’s Implication
W. r. t . Fuzzy control this is the most important
(and most simplest) known in the literature. Its
definition is based on the intersection operation
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 M
M ( ) ( )
( , ) min ( ), ( )
y y
R A B
A B A B
R C A C B
x y x y  
  
   


22. 9. Mamdani...Remarks
i. Also known as control implication, it is better
than the conventional PI controller.
ii. Majority of applications are through Mamdani
Implication.
iii. Sometimes subscript M is also written as c
(for conjunction)
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23. Problem Task
Suppose there is a rule “IF x is A THEN y is B”,
where the meanings of x is A & y is B are given
as:
Determine all the implications for these rule sets
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1 2 3 4
1 2 3
0.1 0.4 0.7 1
0.2 0.5 0.9
A
x x x x
B
y y y
 
    
 
 
   
 

24. Further Reading
1. Jang et. al., “Neuro-Fuzzy & Soft Computing”,
PHI, 1997.
2. Driankov, D. et. al., “An Introduction to Fuzzy
Control”, Narosa, 2001.
3. Mamdani, E. H., “Application of Fuzzy
Algorithm for Control of Simple Dynamic
System”, Proc. IEEE, 121(12), 1974.
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