Part of Lecture series on EE646, Fuzzy Theory & Applications delivered by me during First Semester of M.Tech. Instrumentation & Control, 2012 Z H College of Engg. & Technology, Aligarh Muslim University, Aligarh Reference Books: 1. T. J. Ross, "Fuzzy Logic with Engineering Applications", 2/e, John Wiley & Sons,England, 2004. 2. Lee, K. H., "First Course on Fuzzy Theory & Applications", Springer-Verlag,Berlin, Heidelberg, 2005. 3. D. Driankov, H. Hellendoorn, M. Reinfrank, "An Introduction to Fuzzy Control", Narosa, 2012. Please comment and feel free to ask anything related. Thanks!

1. EE-646
Lecture-7
Fuzzy Relations

2. Fuzzy Relations
• Fuzzy relations map elements of one universe,
X, to those of another universe, Y, through the
Cartesian product of the two universes
• This is also referred to as fuzzy sets defined on
universal sets, which are Cartesian product
• A fuzzy relation is based on the concept that
everything is related to some extent or
unrelated
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3. Fuzzy Relations
• A fuzzy relation is a fuzzy set defined on the
Cartesian product of classical sets
{X1, X2,..., Xn} where n-tuples (x1, x2,..., xn) may
have varying degree of membership
µR(x1, x2,..., xn) within the relation i.e.
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 
 
 1 2
1 2
1 2
1 2, ,...
, ,...
, ,...
, ,...n
R n
n
nX X X
x x x
R X X X
x x x

 

4. Fuzzy Relations
• A fuzzy relation between two sets X & Y is called
binary fuzzy relation & is denoted by:
• A binary relation is referred to as
bipartite graph when X ≠ Y
• The binary relation defined on a single set X is
called directed graph or digraph. This occurs
when X = Y & is denoted by
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 ,R X Y
   2
, orR X X R X
 ,R X Y

5. Fuzzy Relations
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Let X = {x1, x2,..., xn} , Y = {y1, y2,..., ym}. The
fuzzy relation can be expressed by
n × m matrix called Fuzzy Matrix denoted as:
 
     
     
     
1 1 1 2 1
2 1 2 2 2
1 2
, , ... ,
, , ... ,
,
... ... ... ...
, , ... ,
R R R m
R R R m
R n R n R n m
x y x y x y
x y x y x y
R X Y
x y x y x y
  
  
  
 
 
 
  
 
 
 
 ,R X Y

6. Fuzzy Relations
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A fuzzy relation is mapping from Cartesian
space (X, Y) to the interval [0, 1] where the
mapping strength is expressed by the
membership function of the relation for
ordered pairs from the two universes
A fuzzy graph is the graphical representation of
a binary fuzzy relation
 ,R x y

7. Domain & Range
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The domain of the fuzzy relation is the
fuzzy set, having the membership
function as:
The range of the fuzzy relation is the
fuzzy set, having the membership
function as:
   dom max , ,R R
y Y
x x y x X 

  
 ,R X Y
 dom ,R X Y
   Range max , ,R R
x X
y x y y Y 

  
 ,R X Y
 Range ,R X Y

8. Example
Consider a universe X = {x1, x2, x3, x4} & a
binary fuzzy relation R as:
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 
0.2 0 0.5 0
0 0.3 0.7 0.8
,
0.1 0 0.4 0
0 0.6 0 0.1
R X X
 
 
 
 
 
 
x1
x2
x3
x4
x1 x2 x3 x4

9. Example
Domain = {0.5, 0.8, 0.4, 0.6} (Take max on rows)
&
Range = {0.2, 0.6, 0.7, 0.8}
(Take max on columns)
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10. Digraph
See Board
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11. x1
x2
x3
x4
Sagittal Diagram
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x1
x2
x3
x4
X X
0.2
0.5
0.3
0.7
0.8
0.4
0.1
0.1
0.6

12. Operations on Fuzzy Relations
Let be fuzzy relations on the Cartesian
space X × Y. Then the following operations
apply for the membership values for various set
operations:
1. Union:
2. Intersection:
3. Complement:
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~ ~~ ~
( , ) max[ ( , ), ( , )]R S R S
x y x y x y  

~ ~
( , ) 1 ( , )R R
x y x y  
andR S
~ ~~ ~
( , ) min[ ( , ), ( , )]R S R S
x y x y x y  


13. Operations on Fuzzy Relations
4. Containment:
• All properties like commutativity, associativity,
distributivity, involution, idempotency,
De’Morgan’s laws also hold for fuzzy relation
as they do for crisp relations.
• However, the law of excluded middle and law
of contradiction does not hold good for fuzzy
relations (as for fuzzy sets)
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~~ ~ ~
( , ) ( , )R S
R S x y x y   
~ ~ ~ ~ ~ ~
andR R E R R O   

14. Fuzzy Cartesian Product
• Fuzzy relations are in general fuzzy sets
• We can define Cartesian product as a relation
between two or more fuzzy sets
• Let A & B be two fuzzy sets defined on the
universes X & Y , then the Cartesian product
between A & B will result in fuzzy relation
which is contained in full Cartesian product
space
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R

15. Fuzzy Cartesian Product
i. e.
Where, the fuzzy relation has membership
function
The Cartesian product defined by is
implemented in the same fashion as the cross
product of two vectors
Again, the Cartesian product is not the same
operation as the arithmetic product.
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A B R X Y   
R
( , ) ( , ) min[ ( ), ( )]BR A B Ax y x y x y    
A B R 

16. 2D Fuzzy Relation
• In the case of two-dimensional relations
(r = 2), the Cartesian product employs the idea
of pairing of elements among sets, whereas
the arithmetic product uses actual arithmetic
products between elements of sets.
• Each of the fuzzy sets could be thought of as a
vector of membership values; each value is
associated with a particular element in each
set.
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17. 2D Fuzzy Relation
• For example, for a fuzzy set (vector) that has
four elements, hence column vector of size
4×1, and for a fuzzy set (vector)
that has five elements, hence a row vector size
of 1×5, the resulting fuzzy relation, , will be
represented by a matrix of size 4 × 5,
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A
R
B

18. Example
Suppose we have two fuzzy sets, A defined on a universe
of three discrete temperatures, X = {x1, x2, x3} and B
defined on a universe of two discrete pressures, Y =
{y1, y2}, and we want to find the fuzzy Cartesian product
between them. Fuzzy set A could represent the
‘‘ambient’’ temperature and fuzzy set B the ‘‘near
optimum’’ pressure for a certain heat exchanger, and the
Cartesian product might represent the conditions
(temperature–pressure pairs) of the exchanger that are
associated with ‘‘efficient’’ operations.
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19. Example
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Let
A can be represented as a column vector of size
3×1 and B can be represented by a row vector of
1×2. Then the fuzzy Cartesian product results in
a fuzzy relation (of size 3×2) representing
‘‘efficient’’ conditions
1 2 3 1 2
0.2 0.5 1 0.3 0.9
&A B
x x x y y
   
       
  

20. Example
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1 2
1
2
3
0.2 0.2
0.3 0.5
0.3 0.9
A B R y y
x
x
x
 
 
   
  