Fuzzy Relations
Adriano Joaquim de O Cruz
©2009
NCE/UFRJ
adriano@nce.ufrj.br
Summary





Introduction
Crisp Relations
Operations with Crisp Relations
Fuzzy Relations
Operations with Fuzzy Relations
@2009 Adriano Cruz
NCE e IM - UFRJ
Relations 2
Introduction




Relations are associations between
elements of two or more sets.
If the degree of association is one or
zero there is a crisp association.
Degrees of association can be between
0 and 1 in a fuzzy relation
For example the relation x is greater
than y.
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Relations 3
Functions and Relations

Functions and Relations are mappings.

Functions are many to one mappings.

Relations can map many to many.
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Relations 4
Functions
X
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f(X)
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Y
Relations 5
Relations
X
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f(X)
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Y
Relations 6
Cartesian Product

The Cartesian product of two crisp sets
X and Y is defined as
X  Y  {( x, y ) | x  X e y  Y }

For n sets (Xi) the Cartesian product is
defined as
X 1  X 2    X n  {( x1 , x2 , , xn ) | xi  X i , i  1..n}
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Relations 7
Crisp Relations

An relation is a subset of the Cartesian
R( X 1 , X 2 ,, X n )  X 1  X 2   X n
product

The Cartesian product can be considered a
relation without restrictions.

A relation is also a set, therefore the basic set
concepts such as union, intersection,
complement, … can be applied.
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Relations 8
Crisp Relations
Characteristic Function

Shows the strength of the relation
between the pairs.
1 ( x, y )  R
 R ( x, y )  
0 ( x, y )  R

Every tuple that belongs to the relation
receives a value 1 and 0 otherwise.
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Relations 10
Binary Relations

A relation between two sets X and Y is
called a binary relation (R(X,Y)).

Binary relations can be defined on a
single set (R(X,X)).

These relations are often referred as
directed graphs or digraphs
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Relations 11
Representing Relations


Sets of ordered tuples.
Consider a family and the relation is cousin of
X  { Beatriz , Clara , Débora , Marco }
R  cousin of
R XX
R  {( Beatriz, Débora), ( Beatriz, Marco),
(Clara, Débora), (Clara, Marco), ( Débora, Beatriz),
( Débora, Clara), ( Marco, Beatriz), ( Marco, Clara)}
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Relations 12
Representing Relations

N-dimensional membership matrices
Cousin
Beatriz
Clara
Débora
Marco
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Beatriz Clara Débora Marco
0
0
1
1
0
0
1
1
1
1
1
1
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0
0
0
0
Relations 13
Representing Relations

Diagrams that display elements as points
and the relations as arrows between
points (Sagittal diagrams).
Beatriz
Beatriz
Clara
Clara
Débora
Débora
Marco
Marco
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Relations 14
Representing Relations

Simple Diagrams.
Beatriz
Débora
Clara
Marco
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Relations 15
Representing Relations

Equations
x, y  X x R y
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Relations 16
Special Relations

Consider a set A={0,1,2} and the
relations shown below on A  A

Identity Relation I = {0,0),(1,1),(2,2)}

Universal Relation
U={(0,0),(0,1),(0,2),(1,0),(1,1),(1,2),(2,0)
,(2,1),(2,2)}
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Relations 17
Relations - Continuous Universes
y  2x
R  {( x, y ) | y  2 x, x  X , y  Y }
1
 R ( x, y )  
0
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y  2x
y  2x
Relations 18
Crisp Relation Properties

Let X and Y be two sub-sets defined on an
universe X.

Consider the elements x and y.

Let S be the Cartesian product X Y .

Let R be a binary relation defined on S.

Let Q be a binary relation defined on X  X.
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Relations 19
Properties of Crisp Binary Relations

Reflexive: Q is reflexive if and only if (x,x)R
for every xX.

Irreflexive: Q is irreflexive if there is at least
one x such as (x,x)R.

Antireflexive: Q is antireflexive if (x,x)  R for
all xX.
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Relations 20
Examples of Reflexive Relations




Is greater/less than or equal to x
Is a subset of
1 2 3 4 5
Divides
1  √ √ √ √
Is equal to
2
 √ √ √
3
4
5
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
√
√

√

Relations 21
Ex of Anti-Reflexive Relations



Is not equal to
Is greater than
Is coprime
x
x>y
1
2
3
4
y
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5
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1 2 3 4 5
X √ √ √ √
X
√
√
√
X
√
√
X
√
X
Relations 22
Properties of Crisp Relations

Symmetric: R is symmetric if and only if
(x,y)R e (y,x)R for all element xX e yY.

Asymmetric: R is asymmetric if there is no
elements xX and yY such as (x,y)R and
(y,x)R.

Antisymmetric: R is antisymmetric if for all
xX and yY, whenever (x,y)R and (y,x)R
then x=y.
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Relations 23
Examples of Symmetric Relations



Is odd … and is odd too
Is married to
Is equal to
1
y
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1 2
√
3 4
√1
5
√2
2
3
x is odd and y is odd too
x
√1
√
√3
√2
√3
√
4
5
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Relations 24
Properties of Crisp Relations

Transitive: R is transitive if for all x,y,z 
X, if (x,y)R and (y,z)  R then (x,z)R.

Antitransitive: R is antitransitive if (x,z)
 R, whenever (x,y)R and (y,z)  R.

Connected: R is connected if for all x,y
 X, if xy then (x,y)R or (y,x)R.
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Relations 25
Examples of Transitive Relations
x




Is greater than
Is subset of
Divisibility
Implies
1
1
2
3
4
5
6
7
8
√
√
√
√
√
√
√
√
2
3
4
y
5
6
x is divisible by y
7
8
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√
√
√
√
√
√
√
√
√
√
√
√
Relations 26
Transitive Relations




The converse of a transitive relation is always
transitive: if “is a subset of” is transitive and “is a
superset of” is its converse then is “a superset of” is
transitive.
The intersection of two transitive relations is always
transitive: if "was born before" and "has the same
first name as" are transitive then "was born before
and has the same first name as" is transitive.
The union of two transitive relations is not always
transitive. For instance "was born before or has the
same first name as" is not generally a transitive
relation.
The complement of a transitive relation is not always
transitive.
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Relations 27
Properties of Crisp Relations

Left Unique: R is left unique when for all
x,y,zX, if (x,z)R and (y,z)R then x=y.

Right Unique: R is unique when for all
x,y,zX, if (x,y)R and (x,z)R then y=z.

Biunique: a relation R which is both left
unique and right unique is called biunique.
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Relations 28
Example Relation = cousin of


The relation is not reflexive because no one is
cousin of himself, therefore it is antireflexive
and irreflexive.
The relation is symmetric because if Beatriz is
cousin of Débora then Débora is cousin of
Beatriz.Therefore cousin of is not asymmetric.
Beatriz
Débora
Clara
Marco
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Relations 29
Example Relation = cousin of

The relation is also not antisymmetric
because it is not reflexive nor
asymmetric.
Beatriz
Débora
Clara
Marco
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Relations 30
Example Relation = cousin of

The relation is not transitive because
Débora is Clara’s cousin and Clara is
Marco’s cousin, but Débora is not a
cousin of Marco.
Beatriz
Débora
Clara
Marco
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Relations 31
Example Relation = cousin of

The relation is not connected because
there are pairs of different elements to
which the relation is not applicable. For
example Marco is no cousin of Débora.
Beatriz
Débora
Clara
Marco
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Relations 32
Example Relation = cousin of

The relation is not left unique because
Beatriz and Clara are different persons and
both are Débora’s cousin.
Beatriz
Débora
Clara
Marco
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Relations 33
Example Relation = cousin of


The relation is not right unique because
Beatriz is a cousin of Débora and Marco
which are different persons.
Beatriz
Débora
Clara
Marco
The relation is neither left unique nor right
unique therefore it is not biunique.
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Relations 34
Crisp Equivalence Relations

A crisp binary relation R(X,X) that is reflexive,
symmetric and transitive is called an
equivalence relation.

The similarity of triangles is an equivalence
relation.

Work at the same building is an equivalence
relation.
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Relations 35
Crisp Equivalence Relations
For each element x X, there is a crisp set
Ax, which contains all elements of X that are
related to x by the equivalence relation R.
 Ax = { y | (x,y)  R(x,y) }
 xR due to reflexivity of R.
 Each member of Ax is related to all the other
members of Ax because R is transitive and
symmetric.
 No element of Ax is related to any element of
X not included in Ax

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Relations 36
Crisp Equivalence Relations Ex







Let X = {1,2,3,…,10}
Let R(X,X) = {(x,y) | x % 3  y % 3}, % is
remainder when divided by 3
This relation is reflexive, symmetric and
transitive therefore is an equivalence relation
on X.
The three equivalence classes are:
A1 = A4 = A7 = A10 = {1,4,7,10}
A2= A5 = A8 = {2,5,8}
A3= A6 = A9 = {3,6,9}
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Relations 37
Crisp Tolerance Relations


Relations that are reflexive and symmetric are
called a compatibility relations or tolerance
relations.
The relation “city x is close to city y” is a
tolerance relation.
– Lisbon is obviously close to itself (reflexive).
– If Lisbon is close to Paris then Paris is close to
Lisbon (symmetric).
– It is not certain that if Lisbon is close to Paris
and Paris is close to Berlin then Lisbon is close
to Berlin (not transitive).
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Relations 38
Partial Order



A Relation that is reflexive, antisymmetric and
transitive is called a partial order relations (≤ ou
≥).
The relation “A is a subet of B (A  B) is a partial
order.
– A  A (reflexive)
– If A  B and B  C then A  C (transitive)
– If A  B and B  A then A = B
(antisymetric)
A Partial Order does not guarantee that all
pairs are comparable
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Relations 39
Strict Order

A Relation that is antireflexive,
antisymmetric and transitive is called a
strict order relation (< ou >).
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Relations 40
Types of Binary Relations
Reflexive Antireflex
Equiv
Quasi
Equiv
Tolerance
Partial
Order
Strict
Order
X
X
Symmet
X
X
X
X
X
X
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Antisymm Transitive
X
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X
X
X
X
Relations 41
Operations with Crisp Relations


Let R and S be two relations on the
Cartesian product XY.
Let O and I be
0 0  0 
0 0  0 

O
   
0 0  0 


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1
1
E

1

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1  1
1  1

  

1  1
Relations 42
Operations with Crisp Relations
Operations


Crisp Relations are basically sets
defined over higher-dimensional
universes, that is Cartesian products
Usual operations such as union,
intersection and so on are also
applicable.
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Relations 44
Operations with Crisp Relations


Let R and S be two relations on the
Cartesian product XY.
Let O and E be
0 0  0 
0 0  0 

O
   
0 0  0 


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1
1
E

1

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1  1
1  1

  

1  1
Relations 45
Properties of Crisp Operations
União :
R  S   RS ( x, y )  max[ R ( x, y ),  S ( x, y )]
Interseção:
R  S   RS ( x, y )  min[ R ( x, y ),  S ( x, y )]
Complemento :
 R ( x, y )  1   R ( x, y )
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Relations 46
Properties of Crisp Operations
Comutativity
A B  B  A
Associativity
A B  B  A
A  ( B  C )  ( A  B)  C
Distributivity
A  ( B  C )  ( A  B)  C
A  ( B  C )  ( A  B)  ( A  C )
A  ( B  C )  ( A  B)  ( A  C )
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Relations 47
Properties of Crisp Operations
Idempotency
A A  A
Identity
A A  A
A  A
A  
A X  X
A X  A
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Relations 48
Properties of Crisp Operations
Exclusion
A A  E
of the middle A  A  
De Morgan A  B  A  B
A B  A B
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Relations 49
Composition of Crisp Relations
R
X
S
Y
Z
T=R°S
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Relations 50
Composition of Crisp Relations
RS 
 y [  R ( x, y)   S ( x, y)]
X Y
  max
  min or
  product
 The operation ° is similar to a
matrix multiplication.
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Relations 51
Example of Composition
x1
y1
z1
x2
y2
z2
x3
y3
z3
x4
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R  X Y
S Y Z
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Relations 52
Example of Composition
1
0
R
0
0

1 0
1 0

0 1
0 1
1 0 0
S  0 0 1 


1 0 0
1
0
RS  
1
1

@2009 Adriano Cruz
0 1
0 1

0 0

0 0
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Relations 53
Example of Composition
x1
z1
x2
z2
x3
z3
1
0
RS  
1
1

0 1
0 1

0 0

0 0
x4
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Relations 54
Fuzzy Relations
Fuzzy Relations



Fuzzy relations (R) map elements from
a set (X) into a set (Y).
The strength of the relations is given by
membership functions that can vary
between 0 and 1.
R:XY[0:1]
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Relations 56
Fuzzy Relations



Let Ai be fuzzy sets.
A fuzzy relation is a subset of the Cartesian
product
R( A1 , A2 ,, An )  A1  A2   An
The Cartesian product can be considered an
relation without restrictions.
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Relations 57
Properties of Fuzzy Relations


Let X and Y two fuzzy subsets defined on an
Universe U.
Let the elements x  X and y  Y with
membership degrees X(x) e Y(y).

Let S be the Cartesian product X Y .

Let R be a fuzzy relation on S.
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Relations 58
Properties of Fuzzy Relations

Properties with similar definitions to crisp
relations:

Reflexive - µR(x,x) = 1

Irreflexive - µR(x,x)  1 for some x

Antireflexive - µR(x,x)  1 for all x

-Reflexive - µR(x,x) >= 
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Relations 59
Properties of Fuzzy Relations




Properties with similar definitions to crisp
relations:
Symmetric - µR(x,y) = µR(y,x)
Assymetric - µR(x,y)  µR(y,x) for some x,y
X
Antisymetric – when µR(x,y) > 0 and
µR(y,x)>0 implies that x = y for all x,y  X
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Relations 60
Properties of Fuzzy Relations



Properties with similar definitions to crisp
relations:
Connected
Left unique, right unique, biunique
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Relations 61
Properties of Fuzzy Relations

Transitive: R is transitive if for all x,y,z we
have that if (x,y)R and (y,z) R then
(x,z)R.
If  R ( xi , x j )  1 and  R ( x j , xk )  2
then  R ( xi , xk )  min( 1 , 2 )
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Relations 62
Fuzzy Similarity Relations



A fuzzy binary relation that is reflexive,
symmetric and transitive is known as a
similarity relation.
An equivalence relation groups
elements that are equivalent.
The similarity can be viewed from two
different points of view.
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Relations 63
Fuzzy Similarity Relations


The similarity can be considered to
group elements into crisp sets whose
members are similar to each other to
some degree.
When the degree is equal to one the
grouping is an equivalence class.
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Relations 64
Fuzzy Similarity Relations


The similarity can also consider the
degree of similarity that the elements of
X have to some specific element x X.
Then for each X a similarity class can
be defined.
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Relations 65
Fuzzy Similarity Relations Ex
a
b
c
d
e
f
g
a
1
.8
0
.4
0
0
0
@2009 Adriano Cruz
b
.8
1
0
.4
0
0
0
c
0
0
1
0
1
.9
.5
d
.4
.4
0
1
0
0
0
NCE e IM - UFRJ
e
0
0
1
0
1
.9
.5
f
0
0
.9
0
.9
1
.5
g
0
0
.5
0
.5
.5
1
Relations 66
Fuzzy Similarity Relations Ex



X = {a, b, c, d, e, f, g}
Level set = { 0, .4, .5, .8, .9, 1}
Five nested partitions
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Relations 67
Fuzzy Similarity Relations Ex
=.4
a
b d
c
e
f
g
g
=.5
a
b
d
c
e
f
=.8
a
b
d
c
e
f
g
f
g
=.9
a
b
d
c
e
=1
a
b
d
c
e
@2009 Adriano Cruz
NCE e IM - UFRJ
f
g
Relations 68
Fuzzy Tolerance Relations


Relations that are reflexive and symmetric
are called tolerance relations.
The fuzzy relation “The city x is close to
the city y” is a tolerance relation.
@2009 Adriano Cruz
NCE e IM - UFRJ
Relations 69
Orderings
Ordering Characteristics


Similarity and tolerance are
characterized by symmetry.
Ordering relations require asymmetry
(or antisymmetry) and transitivity.
@2009 Adriano Cruz
NCE e IM - UFRJ
Relations 71
Partial Ordering




A crisp binary relation R(X,X) that is reflexive,
antisymmetric and transitive is called a partial
ordering.
The symbol  is suggestive of the properties
of this relation
x  y denotes (x,y)  R and x precedes y
A partial ordering does not guarantee
(antisymmetric x  y, but y may not  x) that
all pairs of elements x, y in X are comparable
(x  y or y  x)
@2009 Adriano Cruz
NCE e IM - UFRJ
Relations 72
Strict Ordering


A crisp binary relation R(X,X) that is
antireflexive, antisymmetric and
transitive is called a strict ordering.
Therefore (x,x)  R,
@2009 Adriano Cruz
NCE e IM - UFRJ
Relations 73
Operations with Fuzzy Relations
Operations with Fuzzy Relations


Let R and S be two fuzzy relations on
the Cartesian Product XY.
Let the relations
0 0  0 
0 0  0 

O
   
0 0  0 


@2009 Adriano Cruz
1
1
E

1

NCE e IM - UFRJ
1  1
1  1

  

1  1
Relations 75
Operations with Fuzzy Relations
Union :
R  S   R  S ( x, y )  max[  R ( x, y ),  S ( x, y )]
Intersecti on :
R  S   R  S ( x, y )  min[  R ( x, y ),  S ( x, y )]
Complement :
 R ( x, y )  1   R ( x, y )
@2009 Adriano Cruz
NCE e IM - UFRJ
Relations 76
Properties of Operations
Comutativity
A B  B  A
Associativity
A B  B  A
A  ( B  C )  ( A  B)  C
Distributivity
A  ( B  C )  ( A  B)  C
A  ( B  C )  ( A  B)  ( A  C )
A  ( B  C )  ( A  B)  ( A  C )
@2009 Adriano Cruz
NCE e IM - UFRJ
Relations 77
Properties of Operations
Idempotence
A A  A
Identity
A A  A
A  A
A  
A X  X
A X  A
@2009 Adriano Cruz
NCE e IM - UFRJ
Relations 78
Properties of Operations
Exclusion
of Middle
A A  E
A A  
De Morgan A  B  A  B
A B  A B
@2009 Adriano Cruz
NCE e IM - UFRJ
Relations 79
Composition of Fuzzy Relations
R
X
S
Y
Z
T=R°S
@2009 Adriano Cruz
NCE e IM - UFRJ
Relations 80
Composition of Fuzzy Relations
RS 
 y [  R ( x, y)   S ( x, y)]
X Y
  max
  min or
  product
 The operation ° is similar to a
matrix multiplication.
@2009 Adriano Cruz
NCE e IM - UFRJ
Relations 81
Example of Fuzzy Composition
x1
x2
1.0
0.8
0.9
0.9
y1
z1
0.7
y2
z2
0.8
x3
0.8
1.0
x4
@2009 Adriano Cruz
y3
z3
R  X Y
S Y Z
NCE e IM - UFRJ
Relations 82
Example of Fuzzy Composition
1 0.8 0 
0 0.9 0 

R
0 0 0.8
0 0 1.0 


0.9 0 0 
S   0 0 0.8


0.7 0 0 
 R ( x1 , z1 )  [(1  0.9)  (0.8  0)  (0  0.7)]
0.9  0  0
 000
RS  
0  0  0.7
0  0  0.7

@2009 Adriano Cruz
0  0  0 0  0.8  0
0  0  0 0  0.8  0

000 000 

000 000 
NCE e IM - UFRJ
Relations 83
Example of Fuzzy Composition
0.9
z1
x1
0.8
x2
0.8
z2
0.7
x3
z3
0.9
0
RS  
 0. 7
 0. 7

0 0.8
0 0.8

0 0

0 0
0.7
x4
@2009 Adriano Cruz
NCE e IM - UFRJ
Relations 84
Relation Example
Relation Definition




Consider a relation that express the
relation petite in terms of height and
weight of a female
Consider the range of these variables
as
Height = [1.51, 1.54, 1.57, …, 1.69]
Weight = [40.8, 43.1, 45.4, 47.6, 49.9,
52.2, 54.4, 56.7]
@2009 Adriano Cruz
NCE e IM - UFRJ
Relations 86
Relation Matrix
40.8
43.1
45.4
47.6
49.9
52.2
54.4
56.7
1.51
1
1
1
1
1
1
0.5
0.2
1.54
1
1
1
1
1
0.9
0.3
0.1
1.57
1
1
1
1
1
0.7
0.1
0
1.60
1
1
1
1
0.5
0.3
0
0
1.63
0.8
0.6
0.4
0.2
0
0
0
0
1.66
0.6
0.4
0.4
0
0
0
0
0
1.69
0
0
0
0
0
0
0
0
@2009 Adriano Cruz
NCE e IM - UFRJ
Relations 87
Questions?

What is the degree that a female with a
specific height and a specific weight is
considered to be petit?
– Relation is equivalent to the membership function
of a multidimensional fuzzy set.

What is the possibility that a petit person has
a specific of height and weight measures?
– Relation is the possibility distribution assigned to
a petit person whose height and weight are
unknown.
@2009 Adriano Cruz
NCE e IM - UFRJ
Relations 88
Other question?



Given a two-dimensional relation and
the possible values of one variable,
infer the possible values of the other
variable.
What is the possible weight of a person
who is about 1.63?
Is it possible to this person to weigh
49.9?
@2009 Adriano Cruz
NCE e IM - UFRJ
Relations 89
Answering the question





What is the possibility that the person is 1.51
given that she is about 1.63?
What is the possibility that the person is 1.51
and weighs 49.9?
If both answers are positive then 49.9 is a
possible weigh.
Continue and ask: what is the possibility that
the person is 1.54?
What is the possibility that she weighs 49.9?
@2009 Adriano Cruz
NCE e IM - UFRJ
Relations 90
Rule of inference






(Possible-height(1.51)  Petite (1.51,40.8))
(Possible-height(1.54)  Petite (1.54,40.8))
…
(Possible-height(1.69)  Petite (1.69,40.8))
Possible-weight(40.8)
(Possible-height(1.51)  Petite (1.51,43.1))
…
@2009 Adriano Cruz
NCE e IM - UFRJ


<=>

Relations 91
Rule of inference

weight ( x )



( w j )   Height ( x ) (hi )  Petite(hi , w j )
hi

Composition max-min
Composition max-prod
@2009 Adriano Cruz
NCE e IM - UFRJ
Relations 92