Slovak University of Technology
Faculty of Material Science and Technology in Trnava
Fuzzy Systems
Introduction to Fuzzy Sets and Systems
Introduction to Fuzzy Sets and Systems



The concept of Fuzzy Logic Fuzzy Sets and Fuzzy Systems was conceived by Zadeh, a professor
at the University of California at Berkley. It is presented not as a control methodology, but as a
way of processing data by allowing partial set membership rather than crisp set membership or
non-membership. This approach to set theory was not applied to control systems until the 70's
due to insufficient small-computer capability prior to that time. Professor Zadeh reasoned that
people do not require precise, numerical information input, and yet they are capable of highly
adaptive control. If feedback controllers could be programmed to accept noisy, imprecise input,
they would be much more effective and perhaps easier to implement.
The word fuzzy has become common knowledge and comes up in every day conversation. This
comes as quite a surprise to those of us who specialize in research into fuzzy systems. Scientific
methodology requires strict logic, but one can say that not much effort goes into verification of
premises and assumptions. The premises and assumptions that sciences and technology worry
so little about are the same axioms in mathematics, and this probably comes about because they
are not logical on the whole. At preset, this problems can only be presented through human
perception and experience. If premises and assumptions are not thoroughly investigated in
technical fields, there is the fear of inviting big mistakes. For example, unexpected accidents in
safety systems, nonsensical conclusions in information systems, automation systems that large
balance all occur when design premises are far from the actual circumstances.
Science and technology do their best to exclude subjectivity, but discovery and invention originate
in right hemisphere activities that are based on subjectivity, and logicizing are no more than
secondary processes for gaining the assent of others. The use of subjectivity is even more
effective during the process of objectification.
A
Some notations of crisp set theory




If A,B are sets, then A is a subset of B ( AB) if xAxB for all xA
If U is an universal set, we denote by P(U) set of all subset of U, P(U)={A;AU}.
P(U) is called potential set of universal se U.
If U is finite and has n elements nN, it is known that P(U) is finite and has 2n
elements.
It is patent that P(U) is a Boolean algebra with respect operations union (),
intersection () and complement of sets.

Some basic (standard) operation set

A B={xU;xA or xB}={xU;xA  xB} (the union of sets)

A  B={xU;xA and xB}=={xU;xA  xB} (the intersection of sets)

Ac={xU;xA } (the complement of the set)

A - B=A\B={xU;xA and xB}={xU;xA  xB} (the different of sets).
AB=(A-B)(B-A)= (A\B)(B\A) (the symmetric different of sets)
AB={(x,y);xA and yB}={(x,y);xA  yB} (Cartesian product of sets ).
If A,B are sets, we call relation any non empty subset R AB. If R is a relation,
then notation (x,y)R is the same as xRy.
Some properties of relations

The relation R is
1.
left-total: if for all x in A there exists a y in B such that xRy (this property, although
sometimes also referred to as total, is different from the definition of total in the next
section).
right-total: if for all y in B there exists an x in A such that xRy.
symmetric, if (x,y)R(y,x)R,
reflexive, if (x,x)R
transitive, if [(x,y)R] [(y,z)R] [(x,z)R]
If R is symmetric, reflexive and transitive then it is relation equivalence.
antisymmetric: if for all x and y in B it holds that if xRy and yRx then x = y. "Greater than or
equal to" is an antisymmetric relation, because if x≥y and y≥x, then x=y.
asymmetric: if for all x and y in A it holds that if xRy then not yRx. "Greater than" is an
asymmetric relation, because if x>y then not y>x.
functional (also called right-definite): for all x in X, and y and z in Y it holds that if xRy and
xRz then y = z.
funkcional is surjective: if for all y in B there exists an x in X such that xRy.
funkcional is injective: if for all x and z in A and y in B it holds that if xRy and zRy then
x = z.
funkcional is bijective: left-total, right-total, functional, and injective.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
Mapping (function) A onto B.
If non empty relation fAB have following properties
1) for all xA there exists yB so (.x,y)f
1, x  A
2) If [(.x,y1)f and (.x,y2)f]y1=y2A.  x   
0, x  A
then f is also called mapping (function) A onto B.
Notations (x,y)f, y=f(x), f:xy are equivalent.
The mapping O:AA ...  AA is n-ary operation.
If n=2 we have binary operation.
If n=1 we have u-nary operation
If O(x,y)=O(y,x) then the binary operation is commutative.
If O(x, O(y,z))= O(O(x,y), z) then the binary operation is associative
Characteristic function of set

If A is a subset of universal set U, then function defined on U as follows
1, x  A
 A x   
0, x  A

Is a characteristic function of subset A.

It is easy to show that P(U) and set of all characteristic functions CH(U) are isomorphic
(as sets).
There exist bijection P(U) onto CH(U) i.e. there exists two maps : P(U) CH(U) and :
CH(U) P(U) defined by (A)=A, (A)={xU; A(x)=1}=A thus CH(U) P (U).



P(U) is a Boolean algebra with respect operation union, intersection and complement. This
means that following eight identities are valid
Propperties of set operations

1) AB=BA, AB=BA (commutavity)

2) (AB)C=A(BC),

3) (AB)C=(AC|(BC), A (BC) = (AB)(AC) (distributivity)

4) AA=A,

5) A(AB)=A,

6) A=A,

7)

8) A  A



AA=A
(idenpotency)
A(AB)=A (absorption)
A=, UU=U,
A A U
,
(AB)C=A(BC) (associativity)
 U
A A  
,
U 
(´)(x)=max{(x), ´(x)}
(´)(x)=min {(x), ´(x)}
´(x)=1-(x)
UA=A
Definition of fuzzy set

Definition 1.1: Definition of fuzzy set: Let U is an universal set and . A
fuzzy set is a pair {U,}. A function  we call the membership function.

The value of membership function is a degree of membership of x as an
element of set.

The membership function is a graphical representation of the magnitude of
participation of each input. It associates a weighting with each of the inputs
that are processed, define functional overlap between inputs, and ultimately
determines an output response.

The rules use the input membership values as weighting factors to
determine their influence on the fuzzy output sets of the final output
conclusion. Once the functions are inferred, scaled, and combined, they are
defuzzified into a crisp output which drives the system. There are different
memberships functions associated with each input and output response.
Example Error Membership Function
Example Membership Function


Figure illustrates the features of the triangular membership function which is
used in this example because of its mathematical simplicity. Other shapes can be
used but the triangular shape lends itself to this illustration.
The degree of membership is determined by plugging the selected input
parameter (error or error-dot) into the horizontal axis and projecting vertically to
the upper boundary of the membership function (s).
Difference between crisp set (a) and
fuzzy set (b)
Some notations of fuzzy set

Let is a fuzzy set. Then

a support of fuzzy set A~is Supp A =;

~
if support of fuzzy set
A

-cut of fuzzy set

-level of fuzzy set

a kernel of fuzzy set
is Ker A =;
~

if Ker A then

a height of fuzzy set

a singleton
of fuzzy set
 x  0,1
~
A
x U ;  A x  0
~
A
is finite
x U ;then
 x    
A
is A =;
is discreet;
x U ;  A x  
is A =;
x U ;  A x  1
A
~is
A
normal else is subnormal;
is
~ ;
A
is the set with one element;
A

if
then the fuzzy set is crisp (conventional
Some notations on fuzzy sets
 A x  0,1
Representation theorem of fuzzy set
In applications of mathematics the useful notation is a number of
elements of set (cardinality of set).
Theorem: Let 01 and A, A are cuts of fuzzy set
A  A.
~
A.
Then
Proof: Let  and A~ is fuzzy with universe U, Then
A =  x U ;  A  x     = x U ;  A x    x U ;    A x   A.
(Representation theorem of fuzzy set): Let
~
A=
(U,A). Then
~
A=
0,1
Theorem: Let
is
~
A
is fuzzy set
{A a ;  0,1 }.
~
A =A;0,1.
Then its membership function
A( x)  sup  0,1 ; x  A 
Example
Example: Let
  2,5  2  pre 0  1
A  
  ,   pre   0
fuzzy set. What is vertical?
Solution: The graph of
system
  2,5  2  pre 0  1
A  
  ,   pre   0
is on fig. For -cuts I valid
 0, pre x  - ,2 
 x - 2, pre x  2,3

 A x    5  x
, pre x  3,5

2
 0, pre x  5,  

is horizontal definition of
Slovak University of Technology
Faculty of Material Science and Technology in Trnava
Fuzzy Systems
Measures on Fuzzy Sets
The cardinality of fuzzy set
In the case of finite crisp (convential) sets the number of elements set A is
A  Card A 
  A ( x)    A ( x)
xU
x A
An extension of this term on fuzzy set is
~
~
Definition: Let A is fuzzy set and its universe is finite then the cardinality fuzzy set A
is
~
Card A =

 A ( x)
xSuppA
It is useful to have some variable(measure) as mean membership of elements of
fuzzy set. Those characteristic is
~
Definition: Let A is fuzzy set and its universe is finite then the relative cardinality
~
fuzzy set A is
  A ( x)
~ xSuppA
card A =

 A ( x)
xSuppA
~
Example: Let A =(0,0.5), (1,0.7), (2,0), (3,0.8), (4,1), (5,0.7), (6,0.1), (7,0), (8,0.4),
~
(9,0.9), (10,0.7) then Card A =0,5+0,7+0.8+1+0,7+0,1+0,4+0,9+0,4=
~
~ Card A 5.5
 0 .5
=5,5 and card A =
=
SuppA 11
~
It means that mean value of membership elements of fuzzy set A is 0.5.
~
In this example is Supp A =0,1,2,4,5,6,8,9,10. If =0.7 then -cut of A is
A0.7= 1,3,4,5,9,10 and for =0.7  -level A0,7= 1,5,10. Its kernel is Ker A= 4.
If universal set infinite, membership function is integrable and Supp A is
measurable then relative cardinality of fuzzy set is defined as
Definition 5: Let membership function is integrable on measurable set Supp A. then
~
the relative cardinality fuzzy set A is
  A ( x)dx
~ SuppA
card A =
 dx
SuppA

 x , x  0,1
is of membership function of fuzzy set. What is
0
,
x

0
,
1


the relative cardinality of the fuzzy set?
Example: Let
 A x   
Solution: Let us compute
1
  A ( x)dx
~ SuppA
card A =
 dx
SuppA


x dx
0
1
 dx
0

2
3
x
3
1
0

2
3
Center of fuzzy set
We often need the representative of object(fuzzy set). It usually is mean value or
center of object. The center is easy interpreted as a representative of object (fuzzy
set).
Definition 6: Let membership function is integrable on measurable set Supp A. he
coordinates of center fuzzy set are
 xi  A ( x)dx
ti 
SuppA
  A ( x)dx
, for i=1,2,…,n
SuppA
 x3
 , x  0,2
Example: Let  A x    8
is of membership function of fuzzy set. What is
 0, x  0,2
the centre of the fuzzy set ?
Solution: Let us compute
2
 xi  A ( x)dx
ti 
SuppA
  A ( x)dx
SuppA

x3
 x 8 dx
0
2

0
 
 
1 5
x
40

1 4
x3
x
dx
32
8
2
0
2
0
4 32 8
 . 
5 16 5
Measure of uncertainty
Fuzzy measure theory considers a number of special classes of measures, each of
which is characterized by a special property. Some of the measures used in this theory
are plausibility and belief measures, fuzzy set, membership function and the classical
probability measures. In the fuzzy measure theory, the conditions are precise, but the
information about an element alone is insufficient to determine which special classes of
measure should be used. The central concept of fuzzy measure theory is fuzzy measure,
which was introduced by Sugeno in 1974.
Fuzzy measure m: F(U)R+ can be considered as generalization of the classical
probability measure. A fuzzy measure m over a set U (the universe of discourse with the
~
subsets A , B~ ,...) satisfies the following conditions when U is finite:
1. m(A)  0.
  
2. m  Ai    m Ai 
 i 1  i 1
Definition of measure of uncertainty
Let F(U) is a set of all fuzzy sets over the universe of discourse U. Then measure of
uncertainty is a function m: F(U)R+ , if
1.
for all A~ , B~ F(U) is m( A~ )+m( B~ )=m( A~  B~ )+m( A~  B~ ),
2. If A(x)0,1for all xU then m( A~ )=0, U.(if fuzzy set is crisp set then measure of
uncertainty is zero).
3. If A(x)=0,5 for all xU then m( B~ )m( A~ )for all B~  F(U)
4. Let A(x) B(x) if B(x)0.5 and A(x) B(x) if B(x) 0.5.
Then
m( A~ )m( B~ ).
Measure of uncertainty of discrete fuzzy set
Hamming’s measure of uncertainty of fuzzy set is
Hm( A) 
  A x   A0.5 x
xSuppA
If we compute max
~ Hm( A) 
A
  A0.5 x  U
xU
Norm Hamming’s measure of uncertainty of fuzzy set is
Hm( A ) 
2Hm( A )
U
Euclid’s measure of uncertainty
Euclid’s measure of uncertainty of fuzzy set is
Eu( A ) 
2
  A x    A0.5 x 
xU
If we compute max
~ Eu ( A) 
A

xU
 A0.5 x  
U
2
Norm Euclid’s measure of uncertainty of fuzzy set is
Eu( A ) 
2Eu( A )
U
Entropic measure of uncertainty of fuzzy set
Entropic measure of uncertainty of fuzzy set is
Ent ( A )     A x  ln  A x   1   A x  ln1   A x 
xU
If we compute
max
~ Ent ( A) 
A

   A0.5 xln  A0.5 x  1   A0.5 xln 1   A0.5 x 
xU
  2 ln 2  1  2  ln 1  2   U ln 2
1
1
1
1
xU
Norm entropic measure of uncertainty of fuzzy set is
Ent( A ) 
Ent( A )
U . ln 2
Where - U number (cardinality of U) of element of U
-  A x  is value of membership function in x,
1
~ .
-  A0.5  x  
is value of characteristic function of A0.5 cut fuzzy set A
2
An example
Let A~ =(-1,0.5),(0,0.2),(5,1),(7.0.9),(10,0.2). Calculate standardized Hamming's, Euclid's
and entropic measure uncertainty of fuzzy set.
Universal set is U=-1,0,5,7,10) and A0.5 cut of A~ is A0.5=x;A(x)0.5=-1,5,7 and
1, x  A0.5
characteristic function is  A0.5  
. Then
0
,
x

A
0.5

Hm( A )    A x    A0.5 x   0.5  1  0.2  0  1  1  0.9  1  0.2  0 
xU
 1.
Hm( A ) 
2Hm( A ) 2.1

 0.4
U
5
An example 2
Hm( A )    A x    A0.5 x   0.5  1  0.2  0  1  1  0.9  1  0.2  0 
xU
 1.
Hm( A ) 
Eu( A ) 
2Hm( A ) 2.1

 0.4
U
5
2
  A x    A0.5 x  
xU

2
0.5  1  0.2  0
2
2
2
 1  1  0.9  1  0.2  0
2

 0 ,34  0 ,58
Eu( A ) 
2Eu( A )
U

2.0 ,58
5
 0.52
Ent ( A )     A x  ln  A x   1   A x  ln1   A x  
xU
 0 ,5 ln 0 ,5  0 ,2 ln 0 ,2  1 ln1  0 ,9 ln 0 ,9  0 ,2 ln 0 ,2 
 1  0 ,5  ln1  0 ,5   1  0 ,2  ln1  0 ,2   1  0 ,5  ln1  0 ,5   1  0 ,9  ln1  0 ,9  
- 1  0 ,2  ln1  0 ,2   1,65
Ent( A ) 
Ent( A )
1,65

 0 ,48
U . ln 2 5 * 0 ,69
Measure of uncertainty of fuzzy set if U=R
However each case employed measure of uncertainty must it satisfy common condition 14 of definition of measure uncertainty of fuzzy set. For instance, let fuzzy set is up
universal real numbers and supporter fuzzy set is interval <a,b> and let membership
function on <and,b> integrable. Let
1   A x  , ak  A x   0,5
g ( x)  1   A  x   
  A x  , inak
Then measure uncertainty is
b
m A   g ( x)dx
a
Ant its standardized (norm) version will be
m A 
2m( A)
ba
Or
b
M  a    g 2 ( x)dx
a
And its standardized (norm) version will be
M  A 
4 f ( A)
b  a 
Demonstrate, that function is measure of uncertainty fuzzy set stands to prove
conditions
1.
2.
3.
4.
it satisfy
m( A~ )+m( B~ )=m( A~  B~ )+m( A~  B~ ),
m( A~ )=0, if A(x)0,1 for all xU
Let A(x)=0,5 for all xU. Then m( B~ )m( A~ ) for all B~  F(U)
Let A(x) B(x) for every x, for which B(x)0.5and let A(x) B(x) for every x, for
which B(x) 0.5
Then
m( A~ )m( B~ )
The condition 1 of definition measure of uncertainty pays because integral is contents
of plane areas, it is measure and every measure it must satisfy. Count
m( A  B)  m A \ ( A  B)   m A  B  mB \ ( A  B)   m A  B  m A  B  
m ( A)
m( B )
 m( A)  m( B)  m A  B
and so
m( A  B )  m( A)  m( B)  m A  B
m( A  B)  m A  B  m( A)  m( B)
m( A  B)  m A  B  m( A)  m( B)
Fuzzy complement. Membership function
A fuzzy set operation is an operation on fuzzy set. These operations are generalization
of crisp set operations. There is more than one possible generalization. The most widely
used operations are called standard fuzzy set operations or elementary operations.
There are three operations: fuzzy complements, fuzzy intersections and fuzzy union.
Standard fuzzy complement is defined by
µc(x)=1- µ(x), for all xU
Membership function of fuzzy intersection
Standard fuzzy intersection is defined by
µAB (x)=min{ µA(x), µB(x)} for all xU
Membership function of fuzzy union
Standard fuzzy union is defined by
µAB (x)= max { µA(x), µB(x)} for all xU
Fuzzy complements
Membership function µ(x) is defined as the degree to which x belongs to. Let c A~
denote a fuzzy complement of A~ of type c. Then cµ(x) is the degree to which x belongs to
c A~ , and the degree to which x does not belong to A~ . (µ(x) is therefore the degree to
which x does not belong to c A~ .)
Axioms for fuzzy complements
The generalization of standard complements of fuzzy set is any operation satisfying
next axioms
Axiom c1. Boundary condition
C(0) = 1 and C(1) = 0
Axiom c2. Monotonicity
For all a, b <0, 1>, if a ≤ b, then C(a) ≥ C(b)
Axiom c3. Continuity
C is continuous function.
Axiom c4. Involutions
C is an involution, which means that C(C (a)) = a for each a <0,1>.
Standard fuzzy complement satisfy axioms c1-c4. We prove it
Axiom c1:
Axiom c2:
Axiom c3:
Axiom c4:
If c(µ(x))= µc(x) =1-µ(x) then µc(0) =1-0=1 and µc(1) =1-1=0
If c(µ(x))= µc(x) =1-µ(x) and a≤b then –a≥-b and 1-a≥1-b
If c(µ(x))= µc(x) =1-µ(x) then y=1-t is elementary and so continue function.
If c(µ(x))= µc(x) =1-µ(x) then (µc)c(x)=1-(1-µ(x))=µ(x)
Fuzzy intersections
The intersection of two fuzzy sets A and B is specified in general by a binary
operation on the unit interval, a function of the form
µ:[0,1]×[0,1] → [0,1].
µAB(x) =µ* [A(x), B(x)] for all x.
Axioms for fuzzy intersection
Axiom i1. Boundary condition
µ* (a, 1) = a
Axiom i2. Monotonicity
b ≤ d implies µ* (a, b) ≤ µ* (a, d)
Axiom i3. Commutability
µ* (a, b) = µ* (b, a)
Axiom i4. Associativity
µ* (a, µ* (b, d)) = µ* (µ* (a, b), d)
Axiom i5. Continuity
iµ*is a continuous function
Axiom i6. Subidempotency
µ* (a, a) ≤ a
Fuzzy intersections. An example














An example of fuzzy fuzzy intersection is
µ*(= µAB(x)= min {µA(x),µB(x)}
To prove that we show that it satisfy axioms i1-i6.
Axiom i1.: µ* (a, 1) = a. Let us compute µ* (a, 1)= µ*(= µAU(x)=
=min {µA(x),1 }= µA(x),
Axiom i2. If a=µA(x), µB(x)=b ≤ d= µC(x) implies µ* (a, b) ≤ µ* (a, d).
Let us compute µ* (a, b)= µ*(= µAB(x)= min {µA(x), µB(x) }≤ min {µA(x),
µC(x) }
Axiom i3: Commutativity µ* (a, b) = µ* (b, a). Let us compute µ* (a, b) =
=min {µA(x), µB(x) }= min {µB(x), µA(x) }=µ* (b, a)
Axiom i4. Associativity µ* (a, µ* (b, d)) = min {µA(x), min {µB(x), µC(x) } }=
=min{min{ µA(x), µB(x)}, µC(x) }= µ* (µ* (a, b), d)
Axiom i5. Continuity: µ*(= µAB(x)= min {µA(x), µB(x)}=min{u,v} is
continues
function
Axiom i6. Subidempotency µ* (a, a) = min {µA(x), µA(x) }= µA(x)≤ µA(x)= a
Fuzzy unions
The union of two fuzzy sets A and B is specified in general by a binary operation on
the unit interval function of the form
u:[0,1]×[0,1] → [0,1].
(A  B)(x) = u[A(x), B(x)] for all x
Axioms for fuzzy union
Axiom u1. Boundary condition
u(a, 0) = a
Axiom u2. Monotonicity
b ≤ d implies u(a, b) ≤ u(a, d)
Axiom u3. Commutativity
u(a, b) = u(b, a)
Axiom u4. Associativity
u(a, u(b, d)) = u(u(a, b), d)
Axiom u5. Continuity
u is a continuous function
An example of fuzzy union operation
An example of fuzzy union operation is
~ ~
µ*( A, B ) = µAB(x)= max{µA(x),µB(x)}
To prove that we show that it satisfy axioms u1-u5.
Axiom u1. Boundary condition
~ ~
u(a, 0)=µ*( A, A) = µAA(x)= max{µA(x),µA(x)}= µA (x)= a
Axiom u2. Monotonicity
b ≤ d implies u(a, b) ≤ u(a, d)
Let a= µA (x),b= µB(x),d= µC(x). Then
u(a, b)=max{ µA(x),µB(x)}≤ max{ µA(x),µC(x)}=u(a,db)
Axiom u3. Commutativity
u(a, b) = max{ µA(x),µB(x)}= max{ µB(x),µA(x)}=u(b, a)
Axiom u4. Associativity
u(a, u(b, d)) = u(u(a, b), d)
u(a, u(b, d)) = max{ µA(x), max{ µB(x),µC(x)}}= max{ max{ µA(x),µB(x)}, µC(x }=
=u(u(a, b), d)
Axiom u5. Continuity
u is a continuous function
Aggregation operations
Aggregation operations on fuzzy sets are operations by which several fuzzy sets
are combined in a desirable way to produce a single fuzzy set.
Aggregation operation on n fuzzy set (n≥2) is defined by a function
v:<0,1>n → <0,1>
Axioms for aggregation operations fuzzy sets
Axiom v1. Boundary condition
v(0, 0, ..., 0) = 0 and v(1, 1, ..., 1) = 1
Axiom v2. Monotonicity
For any pair (a1, a2, ..., an) and <b1, b2, ..., bn> of n-tuples such that
ai, bi  <0,1> for all i  N, if ai ≤ bi for all i N, then
v(a1, a2, ...,an) ≤ v(b1, b2, ..., bn); that v is monotonic increasing in all its arguments.
Axiom v3. Continuity
V is a continuous function.
T-norm
In mathematics, a t-norm (also T-norm or, unabbreviated, triangular norm) is a
kind of binary operations used in the framework of probabilistic spaces and in multi valued
logic, specifically in fuzzy logic. A t-norm generalizes intersection in a lattice and
conjunction in logic. The name triangular norm refers to the fact that in the framework of
probabilistic metric spaces t-norms are used to generalize triangle inequality of ordinary
metric spaces.
Definition
A t-norm is a function T: <0, 1> × <0, 1> → <0, >] which satisfies the following properties:




Commutavity: T(a, b) = T(b, a)
Monotonicity: T(a, b) ≤ T(c, d) if a ≤ c and b ≤ d
Associativity: T(a, T(b, c)) = T(T(a, b), c)
The number 1 acts as identity element: T(a, 1) = a
Since a t-norm is a binary algebraic operation on the interval <0, 1>, infix algebraic
notation is also common, with the t-norm usually denoted by * .
The defining conditions of the t-norm are exactly those of the partially ordered Abelian monoid
on the real unit interval <0, 1> (ordered group). The monoidal operation of any partially
ordered Abelian monoid L is therefore by some authors called a triangular norm on L.
Motivations and applications
of T-norm
T-norms are a generalization of the usual two-valued
logic conjunction, studied by classical logic, for fuzzy logic.
Indeed, the classical Boolean conjunction is both
commutative and associative. The Monotonicity property
ensures that the truth value of conjunction does not
decrease if the truth values of conjuncts increase. The
requirement that 1 be an identity element corresponds to
the interpretation of 1 as true (and consequently 0 as false).
Continuity, which is often required from fuzzy conjunction as
well, expresses the idea that, roughly speaking, very small
changes in truth values of conjuncts should not
macroscopically affect the truth value of their conjunction.
T-norms are also used to construct the intersection of
fuzzy sets or as a basis for aggregation operators.
Slovak University of Technology
Faculty of Material Science and Technology in Trnava
Fuzzy Systems
Classification of t-norms
and conorms
Classification of t-norms
Prominent examples
Minimum t-norm
Tmin x, y   min x, y
also called the Gōdel t-norm, as it is the standard semantics for conjunction in Gōdel
fuzzy logic. Besides that, it occurs in most t-norm based fuzzy logics as the standard
semantics for weak conjunction. It is the point wise largest t-norm. By the minimum t-norm
Zadeh defined intersection on fuzzy sets and in many papers is called standard
operation of intersection fuzzy sets.
Product t-norm:
T prod ( x, y )  x. y
(product of real numbers). Besides other uses, the product t-norm is the standard
semantics for strong conjunction in product fuzzy logic. It is a strict Archimedean t-norm.
Łukasiewicz t-norm
TLuk x, y)  max 0, x  y  1 .
The name comes from the fact that the t-norm is the standard semantics for strong
conjunction in Łukasiewicz fuzzy logic. It is a nilpotent Archimedean t-norm, point wise
smaller than the product t-norm.
Drastic t-norm
if x  1
 y,

TD x, y    x,
if y  1
0, otherwice

The name reflects the fact that the drastic t-norm is the pointwise smallest t-norm. It is a
right-continuous Archimedean t-norm.
Nilpotent minimum
min x, y,
TnM x, y   
 0,
if x  y1
othrwice
is a standard example of a t-norm which is left-continuous, but not continuous. Despite its
name, the nilpotent minimum is not a nilpotent t-norm.
Hamacher product
0,

xy
TH 0 x, y   
,

x

y
xy

if x  y  0
otherwice
is a strict Archimedean t-norm, and an important representative of the parametric classes
of Hamacher t-norms and Schweizer Sklar t-norms.
Properties of t-norms
The drastic t-norm is the pointwise smallest t-norm and the minimum is the
pointwise largest t-norm:
TD x, y   T x, y   Tmin x, y 
for any t-norm and all a, b in [0, 1].
For every t-norm T, the number 0 acts as null
element: T(x, 0) = 0 for all x in <0, 1>.
A t-norm T has zero divisors if and only if it has
nilpotent elements; each nilpotent element of T is also
a zero divisor of T. The set of all nilpotent elements is
an interval <0, x> or <0, x), for some x in <0, 1>.
Properties of continuous t-norms
Although real functions of two variables can be
continuous in each variable without being continuous on
<0, 1>2, this is not the case with t-norms: a t-norm T is
continuous if and only if it is continuous in one variable, i.e., if
and only if the functions fy(x) = T(x, y) are continuous for each
y in <0, 1>. Analogous theorems hold for left- and rightcontinuity of a t-norm.
A continuous t-norm is Archimedean if and only if 0 and
1 are its only idenpotents.
A continuous Archimedean t-norm T is nilpotent if and
only if each x < 1 is a nilpotent element of T. Thus with a
continuous Archimedean t-norm T, either all or none of the
elements of (0, 1) are nilpotent. If it is the case that all
elements in (0, 1) are nilpotent, then the t-norm is isomorphic
to the Łukasiewicz t-norm; i.e., there is a strictly increasing
function f such that
T  x, y   f
1
TLuk  f x , f  y 
For each continuous t-norm, the set of its idempotents is a
closed subset of <0, 1>. Its complement — the set of all
elements which are not idempotent — is therefore a union of
countable many non-overlapping open intervals. The restriction
of the t-norm to any of these intervals (including its endpoints) is
Archimedean, and thus isomorphic either to the Łukasiewicz tnorm or the product t-norm. For such x, y that do not fall into the
same open interval of non-idempotents, the t-norm evaluates to
the minimum of x and y. These conditions actually give a
characterization of continuous t-norms, called the Mostert–
Shields theorem, since every continuous t-norm can in this
way be decomposed, and the described construction always
yields a continuous t-norm.
T-conorms
T-conorms (also called S-norms) are dual to t-norms under the order-reversing
operation which assigns 1 – x to x on [0, 1]. Given a t-norm, the complementary conorm is
defined by
S x, y   1  T (1  x,1  y)
This generalizes De Morgan’s laws..
It follows that a t-conorm satisfies the following conditions, which can be used for an
equivalent axiomatic definition of t-conorms independently of t-norms:




Commutativity: S(a, b) = S(b, a)
Monotonicity: S(a, b) ≤ S(c, d) if a ≤ c and b ≤ d
Associativity: S(a, S(b, c)) = S(S(a, b), c)
Identity element: S(a, 0) = a
T-conorms are used to represent logica disjunction in fuzzy logic and union in fuzzy set
theory..
Examples of t-conorms
Maximum t-conorm
S max x, y   max x, y
dual to the minimum t-norm, is the smallest t-conorm. It is the standard semantics for
disjunction in Gödel fuzzy logic and for weak disjunction in all t-norm based fuzzy logics.
Probabilistic sum and bounded sum
Probabilistic sum
S sum x, y   x  y  x. y
is dual to the product t-norm. In probability theory it expresses the probability of the
union of independent events.. It is also the standard semantics for strong disjunction in
such extensions of product fuyyz logic in which it is definable (e.g., those containing
involutive negation).
Bounded sum
S Luk x, y   min x  y,1
is dual to the Łukasiewicz t-norm. It is the standard semantics for strong disjunction
in Lukasiewicz fuzzy logic.
Drastic t-conorm and Nilpotent
maximum
Drastic t-conorm
 y,

S D x, y    x,
1,

if x  0
if y  0
otherwice
dual to the drastic t-norm, is the largest t-conorm . The function is discontinuous at the
lines 1 > x = 0 and 1 > y = 0.
Nilpotent maximum
max x, y, if x  y1
S nM x, y   
otherwice
1,
dual to the nilpotent minimum:
The function is discontinuous at the line 0 < x = 1 – y < 1.
Einstein sum (compare the velocity-addition formula under special relativity)
S H 2  x, y  
x y
1  xy
is a dual to one of the Hamacher t-norms.
Properties of t-conorms
Many properties of t-conorms can be obtained by dualizing the properties of t-norms, for
example:


For any t-conorm, the number 1 is an annihilating element: S(a, 1) = 1, for any a in
<0, 1>.
Dually to t-norms, all t-conorms are bounded by the maximum and the drastic tconorm:
S max x, y   S x, y   S D x, y 
for any t-conorm S and all x,z in <0, 1>.
Unary operations in fuzzy sets
Operation decreasing vagueness, according to definition of measure uncertainty fuzzy set
~ )m( B
~ ), needs
( let A(x) B(x) if B(x)0.5 and A(x) B(x) if B(x) 0.5 then m( A
increase value of membership function when its value is less than 0,5 and value decrease its
value if it is smaller 0,5. So we have
The operation decreasing contrast
F(U)=  A ;  A : U  0,1 , if it satisfy
is
any
mapping
Int:F(U)
Int x   A x if  A x  0.5 and Int  A x   A x if  A x0.5
Example: Mapping
 2 1
1
 A x  
, if  A x   0.5

Int ( x)   2
2
2

2 A x  , if  A x   0.5

Is function (operation) decreasing contrast. It is evident, because
2 1
1
 A x  
  A x   2  1  A x   1  2 A x 
2
2
If t 2   A  x  , then



2  1 t  1  2t 2  2t 2 
If t<0.5,1>, then t 
1
2



1 

2  1 t  1  0  2 t  1 t 
0
2

 0 and t-1≤0 and Int x   A x, if  A x  0.5 .
If t<0,0.5>, then 2t2≤t2t≤1t≤0.5 is true
F(U),
Graph membership function of fuzzy set and Int function
Function defined by
 2 A  x 3  3 A  x 2 , if  A  x   0.5
Int ( x)  
2
2 A  x  , if  A  x   0.5

is function (operation) c too. To prove it we denote t=A(x), then
 2t 3  3t 2 , if t  0.5
Int ( x)  
2
 2t , if t  0.5
If Int is function decreasing contrast, then if t≥0.5 it must satisfy
-2t3+3t2t-2t(t-1)(t-0.5)0t(1-t)(t-0.5)0t-0.50t0.5
is true.
If 0≤ t≤0.5 it must satisfy
2t2≤t2t2-t≤0t(2t-1)≤02t-1≤0t≤0.5
is true too. So Int is function decreasing contrast.
For extension, restriction, value membership functionl we have operation concentration
and dilatation.
Let Con, Dil  F(U). Then Con is operation concentration if and only i
CON( x )   A ( x )
for all xU.
Dil is operation dilatation if and only i
DIL( x )   A ( x )
for all xU.
Example: Functions
CON ( x)   A ( x) , a1
a
DIL( x)   A ( x) ,0a1
a
for all xU are operations concentration and dilatation. For instance
CON( x )   A ( x )
2
DIL( x )   A ( x )
Graphs functions Dil and Con
Slovak University of Technology
Faculty of Material Science and Technology in Trnava
Fuzzy Systems
Fuzzy relations
Fuzzy relations
Definition: (classical n-ary relation) Let X1,...,Xn be classical(crisp) sets. The
subsets of the Cartesian product X1 ×···× Xn are called n-ary relation. If X1
= ··· = Xn and R Un then R is called an n-ary relation (operation) in U.
Let R be a binary relation in R. Then the characteristic function of R is
defined as
1, ( x, y )  R
0, ( x, y)  R
 R x, y   
Example: Consider the following relation
x, y   R  x  a, b  y  c, d
1, ( x, y )  a, b  c, d
 R  x, y   
0, ( x, y )  a, b  c, d
Let R be a binary relation in a classical set
X. Then
Graph relation R
Properties of relations
Definition. (reflexivity) R is reflexive if (x,x)  R for all xU.
Definition. (anti-reflexivity) R is anti-reflexive if f (x,x)  R for all xU.
Definition. (symmetricity) R is symmetric if from (x,y)  R  (y,x) R for all
x,yU.
Definition. (anti-symmetricity) R is anti-symmetric if (x,y)  R and (y,x)  R
then x=y for all x,yU.
Definition. (transitivity) R is transitive if (x, y) R and (y,z)R R then (x, z)  R,
for all x,y,zU.
Example. Consider the classical inequality relations on the real line R. It is clear
that ≤ is reflexive, anti-symmetric and transitive, < is anti-reflexive,
antisymmetric and transitive.
Other binary relations are
Definition. (equivalence) R is an equivalence relation if R is reflexive,
symmetric and transitive
Example.
The relation = on natural numbers is equivalence relation.
Definition. (partial order) R is a partial order relation if it is reflexive,
antsymmetric and transitive.
Definition. (total order) R is a total order relation if it is partial order and
for all x,yU (x,y)R or (y,x)R.
Example. Let us consider the binary relation ”subset of”. It is clear that we have
a partial order relation.
The relation ≤ on natural numbers is a total order relation.
Fuzzy relation
Let U and V be nonempty sets.
U ×V.
A fuzzy relation R is a fuzzy subset of
In other words, R F (U × V ),  R : U V  0,1
It is often used equivalence notation  R ( x, y)  R( x, y) .
If U =V then we say that R is a binary fuzzy relation in U.
Let R be a binary fuzzy relation on R. Then R(x,y) is interpreted as the
degree of membership of the ordered pair (x,y) in R.
Example. A simple example of a binary fuzzy relation on
U = {1, 2, 3},
called ”approximately equal” can be defined as
R(1, 1) = R(2, 2) = R(3, 3)=1,R(1, 2) = R(2, 1) = R(2, 3) = R(3, 2)=0.8 ,
R(1, 3) = R(3, 1)=0.3
The example of fuzzy relation
The example. A simple example of a binary fuzzy relation on
U = {1, 2, 3},
called “approximately equal” can be defined as
R(1, 1) = R(2, 2) = R(3, 3)=1,R(1, 2) = R(2, 1) = R(2, 3) = R(3, 2)=0.8 ,
R(1, 3) = R(3, 1)=0.3
 1 0 .8 0 .3 


In matrix notation it can be represented as  0.8 1 0.8 
 0 .3 0 .8 1 


Operations on fuzzy relations
The intersection
Fuzzy relations are very important because they can describe interactions
between variables. Let R and S be two binary fuzzy relations on X × Y .
Definition: The intersection of R and S is defined by
(R  S)(x,y) = min{R(x,y),S(x,y)}.
Note that R : U ×V → <0, 1>, i.e. R the domain of R is the whole Cartesian
product U × V .
Definition: The union of R and S is defined by
(R  S)(x,v) = max{R(x, z),S(x, z)}
Example: Let us define two binary relations


 x1
R = ”x is considerable larger than y”= 
x
 2
x
 3
y3 y 4 

0 .8 0 .1 0 .1 0 .7 
0 0 .8 0 0 

0.9 1 0.7 0.8 
y1
y2
S = ”x is very close to y”=
The intersection of R and S means that ”x is considerable larger than y” and
„is very close to y”.
y1 y 2 y 3 y 4 



x
0
.
4
0
0
.
1
0
.
6
 1

(R  S)(x,y) =min{R(x,y),S(x,y)}= 
x 2 0 0 .4 0 0 


 x 0 .3 0 0 .7 0 .5 
 3

The union of R and S means that ”x is considerable larger than y” or ”x is very
close to y”.


 x1
x
 2
x
 3
y3 y 4 

0 .8 0 0 . 9 0 .7 
0 .9 0 .8 0 . 5 0 .7 

0.9 1 0.8 0.8 
y1
y2
The basic properties of fuzzy relations
We wil now try to give some basic properties of compositions of fuzzy relations
which plays a major role in areas such as fuzzy control, fuzzy diagnosis and fuzzy
expert systems.
1. R  I  I  R  R
2. R  O  O  R  O
3. In general R  S  S  R
4. R m 1  R m  R  R
5. R m  R n  R n  m
 n  R mn
6. R m
7. ( R  S )  T  R  ( S  T )
8. R  ( S  T )  R  S   R  T 
9. R  ( S  T )  R  S   R  T 
10. S  T  R  S   R  T 
Fort inverse relarions
11. R  S c  R c  S c
 c  R
12. R c
13. R  S  R c  S c
R  S c
 Rc  S c
R  S c
 Rc  S c
Let R* I fuzzy equivalence relation and R*(x,y)≥R(x,y) and for any fuzzy
equivalence relation S, S(x,y)≥R*(x,y), then R* is minimum fuzzy equivalence closer of
R.
Example: Let
 0.9 0.2

 0.3 1
R
0.2 0.5

 0.6 0.2

.7 

0.5 0.7 
0.4 0.7 

0.4 0.8 
0.5
What is minimum fuzzy equivalence closer of R?
The minimum fuzzy equivalence closer of R is fuzzy reflexive relation. The fuzzy
relation is reflexive if for all xU R(x,x)=1. The minimum reflexive relation R*R is relation
R*(x,x)=1 and R*(x,y) =R(x,y) for all xy. Hence
 1 0.2 0.5 .7 


0
.
3
1
0
.
5
0
.
7


R*  
0.2 0.5 1 0.7 


 0.6 0.2 0.4 1 


The fuzzy relation is symmetric if for all x,yU R(x,y)=R(y,x). The minimum symmetric
relation R*R is relation R*(x,y)=max {R(x,y),R(z,x)} for all xy. Hence
Hence
1
max 0.2,0.3 max 0.2,0.5 max 0.6,0.7



1
max 0.2,0.5 max 0.2,0.7
*  max 0.2,0.3
R 

max 0.2,0.5 max 0.5,0.5
1
max 0.4,0.7


 max 0.6,0.7 max 0.2,0.7 max 0.4,0.7

1


 1 0 .3 0 . 5 0 .7 


 0 .3 1 0 . 5 0 .7 

0 .5 0 . 5 1 0 .7 


 0 .7 0 .7 0 .7



The minimum fuzzy transitive relation fuzzy closer of R and if U is finite then
R*=Rn-1. Hence
 1 0 .3 0 .5 0 .7   1 0 .3 0 .5 0 .7 

 

0
.
3
1
0
.
5
0
.
7
0
.
3
1
0
.
5
0
.
7




R2  



0 .5 0 .5 1 0 .7
0 .5 0 .5 1 0 .7 

 

 0 .7 0 .7 0 .7 1   0 .7 0 .7 0 .7 1 

 








max
1
,.
3
,.
5
,.
7
max
.
3
,.
3
,.
5
,.
7
max
.
5
,.
3
,.
5
,.
7
max
.
7,.3,.5,.7



 max .3,.3,.5,.7 max .3,1,.5,.7 max .3,.5,.5,.7 max .3,.7,.5,.7


max .5,.3,.5,.7 max .3,.5,.5,.7 max .5,.5,1.7 max .5,.5,.7,.7


 max .7,.3,.5,.7 max .3,.7,.5,.7 max .5,.5,.7,.7 max .7,.7,.7,1 


 1 0 .7 0 .7 0 .7 


0
.
7
1
0
.
7
0
.
7



0 .7 0 .7 1 0 .7 


 0 .7 0 .7 0 .7 1 


 1 0 . 7 0 . 7 0 . 7   1 0 . 3 0 . 5 0 .7 

 

0
.
7
1
0
.
7
0
.
7
0
.
3
1
0
.
5
0
.
7

 

R3  R 2  R  


0 . 7 0 . 7 1 0 . 7   0 . 5 0 . 5 1 0 .7 

 

 0 . 7 0 . 7 0 . 7 1   0 . 7 0 .7 0 .7 1 

 

 max 1,.3,.5,.7 max .3,.3,.5,.7 max .5,.3,.5,.7 max .7,.7,.7,.7


 max .3,.3,.5,.7 max .3,1,.5,.7 max .5,.5,.7,.7 max .7,.7,.7,.7



max .5,.3,.5,.7 max .5,.5,.7,.7 max .5,.5,1.7 max .7,.7,.7,.7


 max .7,.7,.7,.7 max .7,.7,.7,.7 max .7,.7,.7,.7 max .7,.7,.7,1 


 1 0 .7 0 . 7 0 .7 


0
.
7
1
0
.
7
0
.
7



0 .7 0 .7 1 0 .7 


 0 .7 0 .7 0 . 7 1 


If fuzzy relations is not symmetric then for symmetric closer of R pay
R*(x,y)?R(x,y) and R *(x,y)?R(y,x). At first we take R* (x,y)=max{ R(y,x), R(x,y) }. It can
be interesting to take R*(x,y)=min{ R(y,x), R(x,y) }.
Example: Let
 1 0 .2 0 .5 .7 


 0 .3 1 0 . 5 0 .7 
R
0 . 2 0 . 5 1 0 .7 


 0 .6 0 .2 0 .4 1 


Then the first estimation of R* is
 1 0 .2 0 . 2 0 .6 


0
.
2
1
0
.
5
0
.
2


R´ 
0 .2 0 .5 1 0 .4 


 0 .6 0 .2 0 . 4 1 


The minimum fuzzy transitive relation fuzzy closer of R´, f U is finite, is R*=Rn-1.
Hence
 1 0 .2 0 .2 0 .6   1 0 .2 0 .2 0 .6 

 

0
.
2
1
0
.
5
0
.
2
0
.
2
1
0
.
5
0
.
2




R2  


0 .2 0 .5 1 0 .4   0 .2 0 .5 1 0 .4 

 

 0 .6 0 .2 0 .4 1   0 .6 0 .2 0 .4 1 

 

If fuzzy relations is not symmetric then for symmetric closer of R pay
R*(x,y)≥R(x,y) and R*(x,y)≥R(y,x). At first we take R*(x,y)=max{ R(y,x), R(x,y) }. It can
be interesting to take R*(x,y)=min{ R(y,x), R(x,y) }.
Example: Let
 1 0.2 0.5 .7 


 0.3 1 0.5 0.7 
R
0.2 0.5 1 0.7 


 0.6 0.2 0.4 1 


Then the first estimation of R* is
 1 0.2 0.2 0.6 


 0.2 1 0.5 0.2 
R´ 
0.2 0.5 1 0.4 


 0.6 0.2 0.4 1 


The minimum fuzzy transitive relation fuzzy closer of R´, f U is finite, is R*=Rn-1.
Hence
 1 0.2 0.2 0.6   1 0.2 0.2 0.6 

 

0
.
2
1
0
.
5
0
.
2
0
.
2
1
0
.
5
0
.
2




R2  



0.2 0.5 1 0.4
0.2 0.5 1 0.4 

 

 0.6 0.2 0.4 1   0.6 0.2 0.4 1 

 

Projections on axis
Consider a classical relation R.
1, ( x, y )  a, b  c, d
R  x, y   
0, ( x, y )  a, b  c, d
It is clear that projection (or shadow) of R on the X-axis is the closed interval <a,
b> and its projection on the Y -axis is <c,d>.
Definition: If R is a classical relation in U × V then
ΠX = {x U| y V :(x, y)  R}
ΠY = {yV |x U :(x, y)  R}
Where ΠX denotes projection on U and
ΠY denotes projection on V.
Definition: Let R be a fuzzy binary fuzzy relation on U × V . The projection of R on U is
defined as
ΠX(x) = sup{R(x, y) | y V }
and the projection of R on Y is defined as
ΠY (y) = sup{R(x, y) | x U}
Example: Consider the relation


 x1
R = ”x is considerable larger than y”= 
x
 2
x
 3
y3 y 4 

0 .8 0 .1 0 .1 0 .7 
0 0 .8 0 0 

0.9 1 0.7 0.8 
y1
y2
then the projection on X means that
•x1 is assigned the highest membership degree from the tuples (x1,y1), (x1,y2),
(x1,y3), (x1,y4), i.e. ΠX(x1)=1, which is the maximum of the first row.
•x2 is assigned the highest membership degree from the tuples (x2,y1), (x2,y2),
(x2,y3), (x2,y4), i.e. ΠX(x2)=0.8, which is the maximum of the second row.
•x3 is assigned the highest membership degree from the tuples (x3,y1), (x3,y2),
(x3,y3), (x3,y4), i.e. ΠX(x3)=1, which is the maximum of the third row.
Shadows of a fuzzy relation
~
Definition: The membership function of Cartesian product of A F (U) and
~
B F (V) is defined as
~ ~
( A × B )(x,y) = min{A(x),B(y)}.
for all xU and yV.
Slovak University of Technology
Faculty of Material Science and Technology in Trnava
Fuzzy Systems
Cartesian product of fuzzy sets
Cartesian product of fuzzy sets
It is clear that the Cartesian product of two fuzzy sets is a fuzzy relation.
If A and B are normal then ΠY (A × B)= B and ΠX(A × B)= A.
Really,
ΠX(x) = sup{(A × B)(x, y) | y}
= sup{A(x) ∧ B(y) | y} = min{A(x), sup{B(y)}| y}
= min{A(x), 1} = A(x).
~
Definition: The sup-min composition of a fuzzy set C F (U) and a fuzzy relation R F
(U × V ) is defined as
~
( C  R)(y) = sup {min{C(x),R(x, y)}}
xU
for all yV .
~
The composition of a fuzzy set C and a fuzzy relation R can be considered as the
~
shadow of the relation R on the fuzzy set C .
Cartesian product of fuzzy sets.
Example 1
~ ~
Let R = A × B Is fuzzy relation.
~
~
~ ~
~
Observe the following property of composition A  R = A  ( A × B )= A ,
~ ~ ~
~
~
B  R = B  ( A × B )= B .
~
Example: Let C be a fuzzy set in the universe of discourse {1, 2, 3} and let R be a
binary fuzzy relation in {1, 2, 3}. Assume that
 1 0 . 8 0 .3 


~
C ={(1,0.2),(2,1)(3,0.3)} and R=  0.8 1 0.8 
 0,3 0,8 1 


Using the definition of sup-min composition we get
 1 0 . 8 0 .3 


~
C  R=(0.2,1,0.3)  0.8 1 0.8  =(max{min{0.2,1},min{1,0.8},min{0.3,0.3}},
 0,3 0,8 1 


max{min{0.2,0.8},min{1,1},min{0.3,0.8}},max{min{0.2,0.3},min{1,0.8},min{0.3,1}}=
=(0.8,1,0.8).
Example 2
Example: Consider two fuzzy relations
R = ”x is considerable larger than y”=
S = ”y is very close to z” =
z1
z2


 y1 0.4 0.9
y 0
0.4
 2
 y3 0.9 0.5

 y4 0.6 0.7
z3 

0.3 
0 

0.8 

0.5 
Then their composition is
z1
z 2 z3 


y3 y 4  
  y1 0.4 0.9 0.3 
0.8 0.1 0.1 0.7  
 y2 0
0.4 0  



0 0.8 0 0
RS=
  y3 0.9 0.5 0.8 
0.9 1 0.7 0.8  

 y 4 0.6 0.7 0.5 
 max 0.4,0,0.1,0.6 max 0.8,0.1,0.1,0.7 max 0.3,0,0.1,0.5

  max 0,0,0,0
max 0,0.4,0,0
max 0,0,0,0
 max 0.4,0,0.7,0.6 max 0.9,0.4,0.5,0.7 max 0.3,0,0.7,0.5



 x1
x
 2
x
 3
y1
y2





 0 .6 0 .8 0 .5 


 0 0 .4 0 
 0 .7 0 .9 0 .7 


sup-product composition of fuzzy relations
Definition: (sup-product composition of fuzzy relations) Let R F (U × V ) and S F
(V × T). The sup-product composition of R and S, denoted by RS is defined as
(R S)(x,z) = sup Rx, y .S  y, z 
yV
It is clear that R S is a binary fuzzy relation in U×T.
Example: Consider two fuzzy relations
R = ”x is considerable larger than y”=
S = ”y is very close to z” =
Then their sup-product composition is
z1

y1 y 2 y3 y 4  


  y1 0.4
 x1 0.8 0.1 0.1 0.7  
 y2 0
x
0 0.8 0 0  
RS=  2
  y3 0.9
 x 0.9 1 0.7 0.8 
 3

 y 4 0.6
z2
0.9
0.4
0.5
0.7
z1
z2


 y1 0.4 0.9
y 0
0.4
 2
 y3 0.9 0.5

 y4 0.6 0.7
z3 

0.3 
0 

0.8 

0.5 
z3 

0.3 
0 

0.8 

0.5 
 max 0.32,0,0.09,0.42 max 0.72,0.04,0.5,0.49 max 0.24,0,0.08,0.35

  max 0,0,0,0
max 0,0.72,0,0
max 0,0,0,0
 max 0.36,0,0.63,0.48 max 0.81,0.4,0.35,0.56 max 0.27,0,0.56,0.4

=
 0.42 0.72 0.35 


0.72
0 
 0
 0.63 0.81 0.56 







If possible to define composition of fuzzy relations in another manner.
For instance, operator max we can replace any t-conorm and min any t-norm.
Fuzzy relation is
Reflexive if R(x,x)=1 for all xU.
Symmetric if R(x,y)=R(y,x) for all (x,y)R
Transitive if
R(x, y)  sup R( x, z ).R( z, y)
zU
Total if for all xU R(x,y) >0 or R(y,x)>0.
Anti symmetric if R(x,y) >0 and R(y,x)>0 implies x=z.
Strongly fuzzy transitive if
for all (x,y)R
It is clear there exist a fuzzy transitive relations R* that R* is strongly
transitive and R*(x,y)≥R(x,y)(for example R*(x,y)=1).
Let R* is strongly transitive relations and R*(x,y)≥R(x,y) and for any
strongly transitive transitive relation S,S(x,y)≥R(x,y) S(x,y)≥R*(x,y), then R* is
fuzzy transitive closer of R.
If U is reflexive and has n elements, then R n 1  R  R  ...  R is transitive
n 1
closer of R.
Example
Let
 1 0.2 0.5 .7   1 0.2 0.5 .7 

 

0
.
3
1
0
.
5
0
.
7
0
.
3
1
0
.
5
0
.
7

 

R2  


0.2 0.5 1 0.7   0.2 0.5 1 0.7 

 

 0.6 0.2 0.4 1   0.6 0.2 0.4 1 

 

 max 1,.2,.2,.6 max .2,.2,.5,.2 max .5,.2,.5,.4 max .7,.2,.5,.7










max
.
3
,.
3
,.
2
,.
6
max
.
2
,
1
,.
5
,.
2
max
.
3
,.
5
,.
5
,.
4
max
.
3
,.
7
,.
5
,.
7




max .2,.3,.2,.6 max .2,.5,.5,.4 max .2,.5,1.4 max .2,.5,.7,.7


 max .6,.2,.2,.6 max .2.2,.4,.2 max .5,.2,.4,.4 max .6,.2,.4,1 


 1 0.5 0.5 0.7 


 0.6 1 0.5 0.7 

0.6 0.5 1 0.7 


 0.6 0.4 0.5 1 


 1 0.5 0.5 0.7   1 0.2 0.5 .7 

 

0
.
6
1
0
.
5
0
.
7
0
.
3
1
0
.
5
0
.
7

 

R3  R 2  R  


0.6 0.5 1 0.7   0.2 0.5 1 0.7 

 

 0.6 0.4 0.5 1   0.6 0.2 0.4 1 

 

 max 1,.3,.2,.6 max .2,.5,.5,.2 max .5,.5,.5,.4 max .7,.5,.5,.7










max
.
6
,.
3
,.
2
,.
6
max
.
2
,
1
,.
5
,.
2
max
.
5
,.
5
,.
5
,.
4
max
.
6
,.
7
,.
5
,.
7




max .6,.3,.2,.6 max .2,.5,.5,.2 max .5,.5,1.4 max .6,.5,.7,.7


 max .6,.3,.2,.6 max .2.4,.5,.2 max .5,.4,.5,.4 max .6,.4,.5,1 


 1 0.5 0.5 0.7 


0
.
6
1
0
.
5
0
.
7



0.6 0.5 1 0.7 


 0.6 0.5 0.5 1 


 1 0 .5 0 .7 


0  is reflexive(R(x,x)=1 for all x) and
Example: The relation R   0.5 1
 0 .7 0
1 

symmetric(R(1,2)=R(2,1)=0.5, R(1,3)=R(3,1)=0.7, R(2,3)=R(3,2)=0) and so is is fuzzy
similarity reletion.
The converse fuzzy relation is usually denoted as Rc is defined as
Rc (x,y)=R(y,x)
For all x,yU
Identity relation
I(x,x)=1 for all xU
I(x,y)=0 for all xyU
Zero relation
o(x,y)=0 for all x,yU
Universe relation
Example: The following are examples of these relations
 1 0.5 0.7 
 1 0.2 0.1



c 
R   0.2 1
0   R   0.5 1
0 
 0.1 0
 0.7 0
1 
1 


 1 0.5 0.7 


R   0.5 1
0 
 0.7 0
1 

1 1 1
 0 0 0




O   0 0 0  U  1 1 1
1 1 1
 0 0 0




Let R* is reflexive, symmetric and is strongly fuzzy transitive relation
then R* is fuzzy similarity relation often called fuzzy equivalence relation.
If fuzzy relations is not symmetric then for symmetric closer of R pay
R*(x,y)≥R(x,y) and R*(x,y)≥R(y,x). At first we take R*(x,y)=max{ R(y,x), R(x,y) }.
It can be interesting to take R*(x,y)=min{ R(y,x), R(x,y) }.
Example: Let
 1 0.2 0.5 .7 


 0.3 1 0.5 0.7 
R
0.2 0.5 1 0.7 


 0.6 0.2 0.4 1 


Then the first estimation of R* is
 1 0.2 0.2 0.6 


 0.2 1 0.5 0.2 
R´ 
0.2 0.5 1 0.4 


 0.6 0.2 0.4 1 


The minimum fuzzy transitive relation fuzzy closer of R´, f U is finite, is R*=Rn-1.
Hence
 1 0.2 0.2 0.6   1 0.2 0.2 0.6 

 

0
.
2
1
0
.
5
0
.
2
0
.
2
1
0
.
5
0
.
2

 

R2  



0.2 0.5 1 0.4
0.2 0.5 1 0.4 

 

 0.6 0.2 0.4 1   0.6 0.2 0.4 1 

 

 max 1,.2,.2,.6 max .2,.2,.2,.2 max .2,.2,.2,.4 max .6,.2,.2,.6


 max .2,.2,.2,.2 max .2,1,.5,.2 max .2,.5,.5,.2 max .2,.2,.4,.4


max .2,.2,.2,.4 max .2,.5,.5,.2 max .2,.5,1.4 max .2,.2,.4,.4


 max .6,.2,.2,.6 max .2,.2,.4,.4 max .2,.2,.4,.4 max .6,.2,.4,1 


 1 0.2 0.4 0.6 


0
.
2
1
0
.
5
0
.
4



0.4 0.5 1 0.4 


 0.6 0.4 0.4 1 


 1 0.2 0.4 0.6   1 0.2 0.2 0.6 

 

0
.
2
1
0
.
5
0
.
4
0
.
2
1
0
.
5
0
.
2

 

R3  



0.4 0.5 1 0.4
0.2 0.5 1 0.4 

 

 0.6 0.4 0.4 1   0.6 0.2 0.4 1 

 

 1 0.2 0.4 0.6 


0
.
2
1
0
.
5
0
.
4



0.4 0.5 1 0.4 


 0.6 0.4 0.4 1 


T-indistinguishability relation
Definition. T-indistinguishability relation E is a reflexive and symmetric
fuzzy relation such that
T(E(x,y),E(y,z))≤E(x,z)
for all x,y,zU.
Definition. A S-pseudometric m is a mapping m:UU<0,1> such that
-m(x,x)=0
-m(x,y)=m(y,x)
S(m(x,y),m(y,z))≥m(x,z)
for all x,y,zU.
There is a close relation between T-indistinguishability relations and
S- pseudometrics as is shown in the following theorem:
Theorem. Let E be a T- indistinguishability relation and let  be a continous
order-reversing bijection from <0,1> to <0,1>. Then
mE(x,y)=(E(x,y))
is a S-pseudometric.
To be more concrete, in order to apply the transitive closure method to construct a
similarity relation and, in general, a fuzzy T-transitive relation, a reflexive and symmetric
fuzzy relation has to be used as a starting point. In others words, an index of similarity
relating each couple of elements in the sample set has to be given: each two elements
should be matched, in some way, and then the method is applied to obtain either a
similarity or dissimilarity measure. At this point, the first arising question is the following:
Does it mean that, for instance, from a single criterion, or from the matching of all
elements to one given, no similarity measure can be given? The obvious negative answer
can be stated by assuming that as a result of the single criterion evaluation or the
matching-to-one process, a function
h:U<0,1>
is given, h(x) representing the degree to which x fits the given conditions. In this
assumption it is easy to check that
m(x,y)=h(x)-h(y)
is a pseudo-distance on U. It is also quite obvious, that
E(x,y)=1-m(x,y)
is a likeness relation on U . It is the measure of similarity between the element y , and
any perfect prototype.
For a long time, the only available methods to build up fuzzy transitive
relations have been the transitive closure and related methods. As it has been
pointed out repeteadly, these methods carry on a number of major problems,
like the requirements of both storage and computer-time and, in spite of this, no
one is satisfied with the results they yield, because there is no way to control
the distorsion that its application produces on the data sample, so that the
transitive closure methods do not fit the desiderata of having a method to
specify a similarity measure which matches with the data.
To be more concrete, in order to apply the transitive closure method to
construct a similarity relation and, in general, a fuzzy T-transitive relation, a
reflexive and symmetric fuzzy relation has to be used as a starting point. In
others words, an index of similarity relating each couple of elements in the
sample set has to be given: each two elements should be matched, in some
way, and then the method is applied to obtain either a similarity or dissimilarity
measure. At this point, the first arising question is the following: Does it mean
that, for instance, from a single criterion, or from the matching of all elements to
one given, no similarity measure can be given? The obvious negative answer
can be stated by assuming that as a result of the single criterion evaluation or
the matching-to-one process, a function
h:U<0,1>
is given, h(x) representing the degree to which x fits the given conditions. In this
assumption it is easy to check that
m(x,y)=h(x)-h(y)
is a pseudo-distance on U. It is also quite obvious, that
E(x,y)=1-m(x,y)
is a likeness relation on U . It is the measure of similarity between the element
y , and any perfect prototype.
Such a construction can be extended in order to get T-transitive fuzzy
relations for any t-norm. If T* stands for the quasi-inverse of the t-norm T , i.e.
then it is also easy to check that
E( x, y)  T *max hx , h y min hx , h y 
is a T-fuzzy transitive relation, such that
h( x)  E ( x, x0 )
for any x0  h 11 . Thus, for instance,
min hx , h y , hx   h y 
E ( x, y )  
1, hx   h y 

is a the similarity relation induced by h , i.e. E is min-transitive. On its own part,
min hx , h y 
E ( x, y ) 
max hx , h y 
is a probabilistic relation, i.e. transitive with respect to the t-norm T(a,b)=a.b and
m(x,y)=1-E(x,y)
is a generalized pseudo-metric with respect to the t-conorm s(a,b)=a+b-a.b.
Summing up, the above considerations show what to do in order to obtain a similarity
measure which matches to the data from a single symmetrica evaluation of the degrees
similarity in the sample set. Next, suppose that several criteria or prototypes are given in
the form of a family of functions h j : U  0,1 in this case the most natural procedure
seems, first, to get the similarity measure –in the form of a fuzzy transitive relation for a
fixed t-norm T – associated with each hj , Ej , and then to take as the degree of the
similarity of two elements, E(x,y), the minimum of all the degrees E j(x,y), which, as it is
easy to check, is also a T-transitive relation. Obviously, there are other ways to combine
fuzzy transitive relations which also preserve the transitive character of the relation. , any
reflexive, symmetric and T –transitive fuzzy relation on a set X is generated by a family o
fuzzy subsets of the given set through the procedure described in this section. In
(Valverde and Ovchinnikov, 1986) it has been shown that the above representation also
holds for left-continuous T –norms, this fact is specially interesting when the minimal T –
norm Z is considered. As it is known, this T –norm is defined by
min x, y, if max x, y  1
Q( x, y)  
1, if max x, y  1

Theorem. Let U be nonempty universal set, S a continuous t-conorm and m a
mapping UU into <0,1>. Then m is pseudometric if, and only if there exist a
 
family h j n , such that
j 1
 

m( x, y)  sup ms j  h j x , h j  y 
j
For some continuous and order reversing bijection  on the unit interval.
In other words, any S-pseudometric on a given set U comes from a
family of fuzzy subsets of the given set. So that, in the case of ordinary
(bounded) metrics, the corresponding S- metric is m( x, y)  x  y . That is, once
a “distance” on the unit interval is fixed, this distance is carried to the given set
U through the fuzzy subsets of U. Let it be noticed that such procedure is
implicitely used in order to associate a likenes relation to a fuzzy partition. As it
is known, at the very ®, a fuzzy partition of a set U was defined as a finite family
of fuzzy subsets ui  of U such that
 ui ( x)  1, for any I and  ui ( x)0 , for any xU.
xU
i
Definition: A function h from U to <0,1> is termed a generator of given T
indistinguisability relation E , if Eh≥E,, HE will denote the set of all generators of
E.
The next definition will play as important role in order to give a more
convenient characterization of the generators of a T-indistinguishability relation
E. It follows immediately from the representation theorem that, given a Tindistin-guishability relation E on U, the set E ( x, y)yU of fuzzy subsets of X is


a generating family of E and will be denoted by h y x 
. The next definition
yU
will play as important role in order to give a more convenient characterization of
the generators of a T-indistinguishability relation E.
Definition. If E be T-indistinguishability relation then E is a map from <0,1>U
into <0,1>U defined by  E h  x   sup T E x, y , h y  for any xU.
yU
If U is a finite set then E is represented by a matrix and  E h may be
understood as the max-T product of E by the column vector representing the
fuzzy set h.
Definition: A function h from U to <0,1> is termed a generator of given T
indistinguisability relation E , if Eh≥E,, HE will denote the set of all generators of
E.
The next definition will play as important role in order to give a more
convenient characterization of the generators of a T-indistinguishability relation
E. It follows immediately from the representation theorem that, given a Tindistin-guishability relation E on U, the set E ( x, y)yU of fuzzy subsets of X is


a generating family of E and will be denoted by h y x 
. The next definition
yU
will play as important role in order to give a more convenient characterization of
the generators of a T-indistinguishability relation E.
Definition. If E be T-indistinguishability relation then E is a map from <0,1>U
into <0,1>U defined by  E h  x   sup T E x, y , h y  for any xU.
yU
If U is a finite set then E is represented by a matrix and  E h may be
understood as the max-T product of E by the column vector representing the
fuzzy set.
Slovak University of Technology
Faculty of Material Science and Technology in Trnava
Fuzzy Systems
Preference relations
Preference relations
More specifically, we define three important relations in A:




A couple of alternatives (a,b) belongs to the strict preference relation P if
and only if the user prefers a to b;
A couple of alternatives (a,b) belongs to the indifference relation I if and
only if the user is indifferent between alternatives a and b;
A couple of alternatives (a,b) belongs to the indifference relation I if and
only if the user is indifferent between alternatives a and b;
A couple of alternatives (a,b) belongs to the incomparability relation J if
and only if the user is unable to compare a and b, for instance caused by
conflicting or insufficient information.
A preference structure on a set of alternatives A is the triplet (P,I,J) of a
binary preference, indifference and incomparability relation in A. However, P, I
and J must satisfy some rather basic additional conditions. For instance, any
couple of alternatives belongs to exactly one of the relations P, Pt (the
transpose of P), I or J. More formally, a preference structure is defined as
follows.
Definition: A preference structure on a set of alternatives A is a triplet (P,I,J) of
binary relations in structures since statements over degrees of and incomparability of preferences are natural and satisfy:
i.
ii.
iii.
iv.
v.
I is reflexive and J is irreflexive;
P is asymmetrical;
I and J are symmetrical;
P I =, P  J =  and I  J = ;
P  Pt  I  J = A 2
Example. Let A=a,b,c and


a
P
b

c

a b c
a



0 1 1
a 1
,
I

b 0
0 0 0



c 0
0 0 0

b c

0 0
,

1 0

0 1 
Then (P,I,J) is preference structure


a
J 
b

c

a b c

0 0 0
0 0 1

0 1 0 
Fuzzy preference relations
Definition. A triplet (P,I,J) of binary fuzzy relations in A is a fuzzy
preference relation on A if and only if
(i)
I is reflexive or (P and J are irreflexive);
(ii)
I is symmetrical or J is symmetrical
(iii)
( (a,b)A2)( P(a,b) + P(b,a) + I(a,b) + J(a,b) =1)).
Let A be a finite set of objects with at least two elements. We interpret the
elements of A as alternatives among which a choice is to be made taking into
account different points of view, e.g. several criteria or the opinion of several
voters. A common practice in such a situation is to associate with each ordered
pair (a, b) of alternatives a number indicating the strength or the credibility of the
proposition “a is at least as good as b”, e.g. the sum of the weights of the
criteria favoring a or the percentage of voters declaring that a is preferred or
indifferent to b. This leads to a fuzzy (large) preference relation on A. In the
area of ELECTRE III is a typical illustration of such a process. A fuzzy (binary)
relation on a set A is a function R associating with each ordered pair of
alternatives (a, b)  A2 an element of 0, 1. Therefore, we define a choice
procedure for fuzzy preference relations (on a set A) as function associating a
nonempty subset of A, the “choice set”, with each fuzzy reflexive binary relation
on A. In this note, we study “choice procedures” instead of the more classical
notion of “choice functions”, i.e. functions associating a choice set with any
subset of A. If a fuzzy relation R is such that R(a, b)  {0, 1}, for all a, b  A, we
say that R is crisp. In this case, we write a R b instead of R(a, b) = 1.
Some properties of choice procedures
It is clear that an ordinal choice procedure does not make use of the
cardinal properties of the numbers R(a, b). Many ordinal choice procedures can
be envisaged. Let us mention one of them that has often been discussed in the
literature and may be seen as a direct extension to the fuzzy case of the
classical notion of the “greatest elements” of a crisp preference relation. Let
R F(A) and, for all a  A, define, using the same notation as in Barrett et al.
(1990), the ‘min in Favor’ score of alternative a letting:
mF a, R   min
R(a, c)
c A \ a
A clearly ordinal choice I defined by CmF R   a  A; mF a, R   mF b, R  for all
bA.
. Let us illustrate the possibility of discontinuities on a simple example involving a
crisp relation and an “almost crisp” one. Consider the relations
R
a
b
c
a
1
0
0
b
1
1
0
c
1
0
1
R´
a
b
c
a
1
0
0
b
1
1
0
c

0
1
where 0 < λ < 1.
It is easy to see that R is crisp and that G® = {a}. Let C be a faithful choice
procedure. We have C® = {a}. Even if C is ordinal, it may happen that a ∉
whatever the value of λ. As a result C®∩C(R´
´ is
arbitrarily “close” to R. Our final axiom is designed to prevent such situations.
 F(A2) if, for all ε ,
there is an integer k such that, for all j ≥ k and all a, b  A, Rj(a,b)-R(a,b) .
A choice procedure C is said to be continuous if, for all RF(A) and all
sequences Rj F(A2) converging to R f(aC(Ri), for all Ri in sequence) aC®
Our definition of continuity implies that an alternative that is always chosen with
fuzzy relations arbitrarily close to a given relation should remain chosen with this
relation. It is not difficult to see that C is continuous.
Fuzzy partial ordered relations
The fuzzy relation is fuzzy partial ordered relation if it satisfy following
conditions
a) is reflexive(R(x,x)=1 for all xU)
b) is symmetric(If R(x,y)0 R(y,x)=0 for all xy)
c) is transitive(R(x,z)supminR(x,y),R(y,z) for all x,zU
 1 0,5 0.6 0.8 


0
1
0
.
7
0
.
9


Example: Fuzzy relation R  
is fuzzy partial ordered relation
0 0
1
1 


0 0
0
1 

Note: Fuzzy relation R is fuzzy partial ordered relation if ad only if its -cut is
patial ordered relation for all 0,1.
Proof: We leave to reader.
Transitivity properties for fuzzy relations
We define the following transitivity conditions
1) Strong stochastic transitivity(S-transitivity)
minR(x,y,R(y,z)0.5R(x,z)maxR(x,y),R(y,z)
2) Moderate stochastic transitivity
minR(x,y,R(y,z)0.5R(x,z) minR(x,y,R(y,z)
3) Weak stochastic transitivity
minR(x,y,R(y,z)0.5R(x,z)0.5
4) -transitivity
minR(x,y,R(y,z)0.5
R(x,z)maxR(x,y,R(y,z)+(1-)minR(x,y,R(y,z)
5) G-transitivity
R(x,z) R(x,y+R(y,z)-1
The
G-transitivity is often called group transitivity because if n elements have
preference R which are linear ordered then
a) R(x,y)0,1
b) R(x,x)=0
c) R(x,y+R(y,z)=1 for xy
d)
R(x,z) R(x,y+R(y,z)-1
It is well-known that from any reflexive binary relation R in a set of alternatives
A, a classical preference structure (P,I,J) can be constructed in the following
way:
t
(i) P = R ∩ coR ;
t
(ii) I = R ∩ R ;
t
(iii) J = co R ∩ coR .
Preference Structures Without Incomparability
Theorem (Roubens and Vincke 1985). A preference structure (P,I,J) on U is a
preference structure (P,I) on U if and only if its large preference relation is
complete.
Two different types of fuzzy preference structures without incomparability
can be distinguished.
Theorem (De Baets and Van de Walle 1995). A fuzzy preference structure
(P,I,J) on U with fuzzy large preference relation R in U is a fuzzy preference
structure (P,I) on A of Type 1 if and only if
(x,y)U2 maxR(x,y),R(y,x)=1
Theorem (De Baets and Van de Walle 1995). A fuzzy preference structure
(P,I,J) on U with large fuzzy preference relation R in A is a fuzzy preference
structure (P,I) on U of Type 2 if and only if
(x,y)U2 R(x,y)+R(y,x)1
In both classes, the following relationship between the fuzzy strict
preference relation P and the fuzzy large preference relation R holds:
(x,y)U2 R(x,y)= 1-R(y,x)
Quasi-order relations and the analysis of preference
relations
A binary (fuzzy) relation R in a universe U is called:
(i) reflexive if and only if xU2 R(x,x)= 1
(ii) a (fuzzy) quasi-order relation in U if and only if it is reflexive and transitive.
Theorem 4 (Fodor and Roubens 1994). Consider a binary fuzzy relation R in a
universe U. R is a fuzzy quasi-order relation in U if and only if for all values of α
(with α belonging to the interval <0,1>) it holds that Rα is a (crisp) quasi-order
relation in U.
The starting point of the analysis is the realization that every row in the matrix
representation of a preference relation is a profile of the preferences a user has
for an alternative compared to all other alternatives. The i-th row of P contains
all preferences of the form P(xi,xj) the number of alternatives. Recall that we can
denote the i-th row of P as the afterset aiP.
Slovak University of Technology
Faculty of Material Science and Technology in Trnava
Fuzzy Systems
Fuzzy functions
Fuzzy functions
One of the fundamental conceptions of mathematics is the function f:AB
. It is nonempty binary relation fAB satisfying conditions
a) xA yB (x,y)f
b) (x1,y)f(x2,y)fx1=x2.
Let F(U) and F(V) are sets of all fuzzy sets on universes U,V. Then a fuzzy
function U in V denoted by f:UV is a map
f: F(U)F(V)
If two fuzzy functions f a g are given
f:UV g:VW
the composition
gf:UW
Examlle: Let U={1,2} and V=R=(-,) and
0, x5

 x  5; x  5,6

f(1)  
7  x; x  6,7

0, x 7
0, x3

 x  3; x  3,4

f(2)  
5  x; x  4,5

0, x5
Then f is fuzzy function from U={1,2} in V=R=(-,).
Definition: The fuzzy function from U in V, denoted by f:UV, is fuzzy subset of
the product UV.
Example: A fuzzy function f ( x, y )  e  x  2 y describes the statement x is
approximatively equal 2y.
0, x1
0, x 2


 x  1, x  1,2
 x  1, x  2,3
~
~


Example: Let A ={R,  A x  
}, B ={R,  B x  
}
3

x
,
x

2
,
3
3

x
,
x

3
,
4




0, x 3
0, x 4
 1   ,3   ;   (0,1
 2   ,4   ;   (0,1


R;   0
R;   0
Z=x+y. Then A = 
B = 




If x 1   ,3    and y 2   ,4    then value membership function is more then
((z) max min{  A (x),  B (y)} and that
x, y 
x  yz
 A (x)   ,  B (y)    x1,  1   , x2,  3   , y1,  2   , y 2,  4  
z1,  x1,  y1,  1    2    3  2 , z 2,  x2,  y 2,  3    4    7  2
0, x3

x3
, x  3,5

~
2
},
C ={R, C z   
7x

, x  5,7
 2

0, x 7
~
~
~
Note: Let A is fuzzy set of U and f is mapping U in V. Then usually projection A onto B
Is fuzzy set with membership function
 B ( y)  max  A x 
y   A x 
~
~
Example: Let A ={(-2,0.4), (-1,0.2),(0,1),(1,0,5), (2,0.8)} and y=x2. Then projection A is
~
B ={(4,max{((-2)2,0.4),
the
fuzzy
set
(22,0.8)},
(1,max{((-1)2,0.2),
(12,0.5)},(0,1)}={(0,1),}1,0.5),(4,0.8)}
0, x1

 x  1, x  1,2
~

Example: Let A ={R,  A x  
} and y=x2
3  x, x  2,3

0, x 3
 1   ,3   ;   (0,1

R;   0
Then A = 
 y  x 2  1   2 , 3   2  y1,  1   2 


 1 
y1,  1,  2  3  y 2, ,

 y

If   0  y1  1, y2  9. If   1  y1  4, y2  4. and  y x   
3 

0, y 1
 1, y  1,4
y , y  4,9
0, y9
FUZZY LOGIC
Fuzzy logic is used in system control and analysis design, because it shortens the
time for engineering development and sometimes, in the case of highly complex systems,
is the only way to solve the problem. Although most of the time we think of "control" as
having to do with controlling a physical system, there is no such limitation in the concept
as initially presented by Dr. Zadeh. Fuzzy logic can apply also to economics, psychology,
marketing, weather forecasting, biology, politics ...... to any large complex system. Fuzzy
logic is not the wave of the future. It is now! There are already hundreds of millions of
dollars of successful, fuzzy logic based commercial products, everything from selffocusing cameras to washing machines that adjust themselves according to how dirty the
clothes are, automobile engine controls, anti-lock braking systems, color film developing
systems, subway control systems and computer programs trading successfully in the
financial markets.
We are all familiar with binary valued logic and set theory. An element belongs to a set of
all possible elements and given any specific subset, it can be said accurately, whether that
element is or is not a member of it.
Unfortunately, not everything can be described using binary valued sets. The
classifications of persons into males and females is easy, but it is problematic to classify
them as being tall or not tall. The set of tall people is far more difficult to define, because
there is no distinct cut-off point at which tall begins. This is not a measurement problem,
and measuring the height of all elements more precisely is
not
helpful.
Such a
proble
m
is
often
distort
ed so that it can be described using the
well known existing methodology. Here, one could simply select a height, e.g. 1.75m,
at which the set tall begins.
Fuzzy logic was suggested by Zadeh as a method for mimicking the ability of
human reasoning using a small number of rules and still producing a smooth output via a
process of interpolation. It forms rules that are based upon multi-valued logic and so
introduced the concept of set membership. With fuzzy logic an element could partially
belong to a set and this is represented by the set membership. For example, a person of
height 1.79m would belong to both tall and not tall sets with a particular degree of
membership. As the height of a person increases the membership grade within the tall set
would increase whilst the membership grade within the not tall set would decrease. The
output of a fuzzy reasoning system would produce similar results for similar inputs. Fuzzy
logic is simply the extension of conventional logic to the case where partial set
membership can exist, rule conditions can be satisfied partially and system outputs are
calculated by interpolation and, therefore, have output smoothness over the equivalent
binary-valued rule base. This property is particularly relevant to control systems.
Rules of propositional calculus
The following table lists some inference rules of propositional calculus The table makes
use of mathematical notation. The following symbols occur in the table:






p  q: p must be true, or q must be true (or both)
p  q: both p and q must be simultaneously true
p  q: p implies q: if p is true then so is q
p  q: p is logically equivalent to q: if either is true/false, then so is the other.
p ├ q: from p infer q (by applying basic inference rules, q can be shown to hold
assuming p (note that this is equivalent to ( ⊢ p → q).
┐p: not p
Basic arguments propositional calculus
Name
Modus Pones
Sequent
[(p → q)  p] ├ q
Modus Tollens
if p then q; not q; therefore not p
(p → q)  ¬q] ⊢ ¬p
[(p → q)  (q → r)] ├ (p
if p then q; if q then r; therefore, if p then r
→ r)
Hypothetical
syllogism
Disjunctive
syllogism
Constructive
dilemma
Description
if p then q; p; therefore q
[(p  q)  ¬p] ├ q
Either p or q; not p; therefore, q
If p then q; and if r then s; but either p or r;
therefore either q or s
Simplification
[(p → q)  (r → s)  (p 
r)] ├ (q s)
[(p → q)  (r → s)  (¬q
 ¬s)] ├ (¬p  ¬r)
(p q) ├ p,q
Conjunction
p, q ├ (p q)
Addition
p ├ (p  q)
[(p → q)  (p → r)] ├ [p
→ (q  r)]
Destructive
dilemma
Composition
De Morgan´s
theorem (1)
¬ (p  q) ├(¬p  ¬q)
If p then q; and if r then s; but either not q or
not s; therefore rather not p or not r
p and q are true; therefore p is true
p and q are true separately; therefore they
are true conjointly
p is true; therefore, for any q, (p or q) is true
If p then q; and if p then r; therefore if p is
true then q and r are true
If it is not true that p and q hold, then at least
either p or q is not true
De Morgan's
Theorem (2)
Commutation(1)
Commutation (2)
¬ (p  q) ├ (¬p  ¬q)
Material
equivalence (2)
(p ↔ q)├ [(p  q) 
(¬q  ¬p)]
[(p  q) → r] ├ [p →
(q → r)]
If it is not true that p or q holds, then p does not
hold and q does not hold
(p or q) is equiv. to (q or p)
(p and q) is equiv. to (q and p)
(p  q) ├ (q  p)
(p q) ├ (q  p)
[p  (q r)]├[(pq)
Association(1)
p or (q or r) is equiv. to (p or q) or r
r]
p and (q and r) is equiv. to (p and q) and r
Association (2)
[p(qr)]├[(p q)r]
(therefore, (p q ∧ r) is unambiguous)
[p(q r)]├[(p
Distribution (1)
p and (q or r) is equiv. to (p and q) or (p and r)
q)(pr)]
[p  (q  r)] ├ [(p 
Distribution (2)
p or (q and r) is equiv. to (p or q) and (p or r)
q)  (p r)]
p ├ ¬¬p
Double negation
p is equivalent to the negation of not p
(p → q) ├ (¬q → ¬p) If p then q is equiv. to if not q then not p
Transposition
Material implication (p → q) ⊢ (¬p ∨ q) If p then q is equiv. to either not p or q
(p↔q)├
Material
(p is equiv. to q) means, (if p is true then q is
equivalence (1)
true) and (if q is true then p is true)
[(p→q)(q→p)]
Exportation
Importation
Tautology
(p is equiv. to q) means, either (p and q are
true) or ( both p and q are false)
from (if p and q are true then r is true) we can
prove (if q is true then r is true, if p is true)
if r is true when q is true, under the condition
[p → (q → r)] ├ [(p 
that p is true, then if p and q are true, r is as
q) → r]
well
p is true is equiv. to p is true or p is false (this
p ├ (p  ¬p)
can be seen as a special case of addition)
The set of logic terms
Terms: The set of terms is recursively defined by the following rules:
1. Any constant is a term.
2. Any variable is a term.
3. Any expression f(t1,...,tn) of n ? 1 arguments (where each argument ti is a term
and f is a function symbol of valence n) is a term.
4. Closure clause: Nothing else is a term.
Well-formed formulas
The set of well-formed formulas (usually called wffs or just formulas) is
recursively defined by the following rules:
1. Simple and complex predicates If P is a relation of valence n ? 1
and the ai are terms then P(a1,an) is well-formed. If equality is
considered part of logic, then (a1 = a2) is well formed. All such
formulas are said to be atomic.
2. Inductive Clause I: If φ is a wff, then ¬φ is a wff.
3. Inductive Clause II: If φ and ψ are wffs, then (φ  ψ) is a wff.
4. Inductive Clause III: If φ is a wff and x is a variable, then ¬x φ is a
wff.
5. Closure Clause: Nothing else is a wff.
Free Variables
Free Variables:
1. Atomic formulas if φ is an Atomic formula then x are free in φ if and only if x
occurs in φ.
2. Inductive Clause I: x is free in ¬φ if and only if x is free in φ.
3. Inductive Clause II: x is free in (φ ? ψ) if and only if x is free in φ or x is free in
ψ.
4. Inductive Clause III: x is free in ? y φ if and only if x is free in φ and x y.
5. Closure Clause: if x is not free in φ then it is bound...
Since ¬ (φ  ¬ψ) is logically equivalent to (φ  ψ), (φ  ψ) is often used as a
short hand. The same principle is behind (φ  ψ) and (φ  ψ). Also  x φ is a short
hand for ¬ y ¬φ. In practice, if P is a relation of valence 2, we often write "a P b"
instead of "P a b"; for example, we write 1 < 2 instead of <(1 2). Similarly if f is a
function of valence 2, we sometimes write "a f b" instead of "f(a b)"; for example, we
write 1 + 2 instead of +(1 2). It is also common to omit some parentheses if this does
not lead to ambiguity.
Sometimes it is useful to say that "P(x) holds for exactly one x", which can be
expressed as! x P(x). This can also be expressed as !x (P(x) y (P(y) ? (x =
y))).
Substitution
If t is a term and φ(x) is a formula possibly containing
x as a free variable, then v φ(t) is defined to be the result
of replacing all free instances of x by t, provided that no
free variable of t becomes bound in this process.
The problem is that the free variable y of t (=y) became
bound when we substituted y for x in φ(x). So to form
φ(y) we must first change the bound variable y of φ to
something else, say z, so that φ(y) is then z z ? y.
Equality
The most common convention for equality is to include the
equality symbol as a primitive logical symbol, and add the
axioms for equality to the axioms for first order logic. The
equality axioms are
1. x = x
2. x = y  f(...,x,...) = f(...,y,...) for any function f
3. x = y  (P(...,x,...) ? P(...,y,...)) for any relation P
(including = itself)
A sentence is defined to be provable in first order logic if
it can be obtained by starting with the axioms of the predicate
calculus and repeatedly applying the inference rules "modus
ponens"
Axioms of basic logic
Let ,,  are formulae then
A1: ()( )()
A2: ()
A3: ()( )
A4:  () ( )
A5a:  ()()
A5b: () ()
A6:  ()(())
A7: 
In classic logic the axiom are
Let ,,  are formulae then
1) ()
2) () ()( )
3) ()()
The membership functions predicates into
fuzzy logic
Kleene-Dienes implication
 R1 ( x , y )  max 1   A x ,  B ( y )  1 x , y 


Lukasiewicz implication
 R2 ( x , y )  min 1,1   A x    B ( y )   2 x , y 


Zadeh implication
 Rm ( x , y )  max min  A x ,  B ( y ) ,1   A x    3 x , y 
 


Stochastic implication
 Rs ( x , y )  min 1,1   A x    B ( y ) A x    4 x , y 

Goguen implication
Goedel implication
Sharp implication
Mamdani implication
Larsen implication

  A x  
 RA ( x , y )  min1,
   5 x , y 
  B ( y ) 
1,  A x    B ( y )
 Rg ( x , y )  n 
   6 x , y 
  B ( y ), inak 
1,  x    B ( y )
 RI ( x , y )   A
  7  x , y 
0 , inak


 RM ( x , y )  min  A x ,  B ( y )   8 x , y 


 RL ( x , y )   A x . B ( y )   9 x , y 
Slovak University of Technology
Faculty of Material Science and Technology in Trnava
Fuzzy Systems
Fuzzy numbers
Fuzzy numbers
A fuzzy set is fuzzy convex set  -cut is convex for all  0,1 .
A fuzzy set is normal if x,  x  1   .
Let U is set of real number. A fuzzy number is a convex normal fuzzy set
whose membership function is at least segmentally continuums. If
x,  x  1  a, b , a  b they also fuzzy number call fuzzy interval.
~
A   R,  
Triangular fuzzy number
A fuzzy number is triangular fuzzy number if its membership function is
0, x a

x  a
, x  a, b

b

a
 x   
cx

, x  b, c
cb

0, x c
or if its horizontal representation is
 a  b  a  , c  c  b  ,   (0,1
A  x1  , x2    
0,   0

A triangular fuzzy number is often called  fuzzy number.
Triangular fuzzy number
Trapezoidal fuzzy number
A fuzzy number is trapezoidal ( or ) if its membership function is
0, x a

xa
, x  a, b

b

a

  x   1, x  b, c
d  x
, x  c, d

d

c


0, x d
or if its horizontal representation is
 a  b  a  , d  d  c  ,   (0,1
A  x1  , x2    
0,   0

Trapezoidal fuzzy number
S and Z fuzzy numbers
A trapezoidal fuzzy number is often expressed as (a,b,c ,d). The triangular number as
(a,b,c). If b=c a trapezoidal fuzzy number is triangular.
Some typical fuzzy numbers
A fuzzy number is positive if cut Afor all 0.
A fuzzy number is negative if cut A for all 0.
If (0)0 the fuzzy number is fuzzy zero.
Representation of fuzzy number
~
Theorem: Let for all   (0,1 cut of fuzzy number A is closed interval. Then there
exist numbers abcd and functions u, v that
u  x , x   , b 

  x    1, x  b, c
 v x , x  c,  

and u(x)=0 for all x (- ,a) and v(x)=0 for all x (d, ).
Proof: It is evident. See picture.
Supremum and infimum of fuzzy number
~
~ ~
Let A, B are fuzzy numbers. Then fuzzy number C is their
~ ~ ~
~
infimum C = A  B if the membership function of C is
  x   supmin  A  x ,  B  y  z  minx, y; x, y  R
~
C
The fuzzy number
is their supremum
~
membership function of C is
~
~ ~
C = A B
  x   supmin  A  x ,  B  y  z  max x, y; x, y  R
if the
Comparable fuzzy numbers
~ ~
~
~ ~ ~
~
~ ~ ~
The fuzzy number A  B ( A is greater B ) if B = A  B or A = A  B .
~ ~
Note A  B if  A x    B x  for all x U.
~ ~
~ ~
~ ~
The fuzzy numbers A, B are comparable if A  B or A  B .
~ ~
~ ~
~ ~
If A  B or A  B is false then A, B are not comparable.
Zadeh's extension principle
You can use fuzzy numbers for fuzzy arithmetic. This can be
done by the application of Zadeh's extension principle.


In the cartesian product of two fuzzy numbers A and B you
take the MINIMUM of the grades of membership of the two
corresponding sub-numbers ai and bi that are operated on,
to determine the grade of membership of the new subnumber ci resulting from that operation.
Then you take the MAXIMUM of the grades of
membership of the subnumbers with the same numerical
value ci to determine the grade of membership of the subnumber ci of the new fuzzy number C. In short it's the
"MAX of MIN's".
Operation addition and difference
The operations on triangular fuzzy numbers are frequently
defined as an operations which result is triangular fuzzy
number.
~
c~  c1 , c2 , c3   a1 , a 2 , a3   b1 , b2 , b3   a~  b
If operation is addition them
~
c~  c1 , c2 , c3   a1 , a 2 , a3   b1 , b2 , b3   a1  b1 , a 2  b2 , a3  b3   a~  b
If operation is difference them
~
c~  c1 , c2 , c3   a1 , a2 , a3   b1 , b2 , b3   a1  b3 , a2  b2 , a3  b1   a~  b
Interval arithmetic
Let
a, b , c, d
are two intervals. Then arithmetical operations arte defined:
Addition:
a, b  c, d  a  c, b  d
Difference:
a, b  c, d  a  c, b  d
Multiply:
a, b  c, d  minac, ad , bc, bd, maxac, ad , bc, bd
Division:
a, b / c, d  a, b
If c,d are positive numbers or negative.
1 1
,
d c
Operation multiplication and division
If operation is multiplication them
~
~
~
c  c1, c2 , c3   a1, a2 , a3   b1, b2 , b3   a1b1, a2b2 , a3b3   a  b
If and only if all fuzzy numbers are positive
If operation is division them
 a1 a2 a3  a~
~
c  c1 , c2 , c3   a1 , a2 , a3  / b1 , b2 , b3    , ,  
 b3 b2 b1  b3
If and only if all fuzzy numbers are positive.
Example
~
Let a~  1,3,4, b  2,3,5 then a   1  2 ,4   , b   2   ,5  2 and
a, b =a1,b1+a2,b2= a1+a2, b1+b2= 1  2 ,4    2   ,5  2 =
= 3  3 ,9  3
a, b =a1,b1-a2,b2= a1-b2, b1-a2= 1  2 ,4    1   ,5  2 =
=  4  4 ,3  2 .
a, b=a1,b1a2,b2=a1,b1.a2,b2=
 1  2 ,4   . 1   ,5  2
= 1  3  2 2 ,20  13  2 2
Extension principle
Let
~ ~ ~
A,B , Z
are fuzzy numbers and A, B, Z their membership functions. Then
membership function for
~ ~ ~
Z =A+B
is defined as
Z ( z )  supminA( x), B( y ) z  x  y
x, y
The membership function for
~ ~ ~
Z = A-B
is defined as
Z ( z )  supminA( x), B( y ) z  x  y
x, y
The membership function for
~ ~ ~
Z =A.B
is defined as
Z ( z )  supminA( x), B( y ) z  x  y
x, y
The membership function for
~ ~ ~ ~
Z =A/ B (B
is non zero number)is defined as
Z ( z )  supminA( x), B( y ) z  x / y
x, y
Slovak University of Technology
Faculty of Material Science and Technology in Trnava
Fuzzy Systems
-cuts and interval arithmetic
Interval arithmetic
Let
a, b , c, d
are two intervals. Then arithmetical operations arte defined:
Addition:
a, b  c, d  a  c, b  d
Difference:
a, b  c, d  a  c, b  d
Multiply:
a, b  c, d  minac, ad , bc, bd, maxac, ad , bc, bd
Division:
a, b / c, d  a, b
If c,d are positive numbers or negative.
1 1
,
d c
Example
Let I1=2,3, I2=-5,-3. What is sum, difference, product and fraction of I1,I2?
I1+I2= 2,3+-5,-3=-3,0.
I1-I2= 2,3--5,-3=2,3+3,5=5,8.
I2-I1= -5,-3-2,3=-5,-3+-3,-2=-8,-5.
I2-I2= -5,-3--5,-3=-5,-3+3,5=-2,2.
I1.I2= 2,3.-5,-3=-15,-9.
1 1
3 2
2
I1 / I 2  2 ,3  / 5 ,3   2 ,3  ,   
,   1, 
3 5
3
5
5
1 1
5 3
5
I 2 / I1  5 ,3  / 2 ,3   5 ,3  ,   
,    ,1 .
3 2
2
3
2
-cuts and interval arithmetic
~
~
Let A is fuzzy number ad all -cuts of A are closed interval. Then its horizontal
representation is set of closed intervals a1 (), a2 () , (0,1 and we can define
operations on fuzzy numbers as operation of interval arithmetic.
Addition of fuzzy numbers
Then membership function for
~ ~ ~
Z = A+B
is defined as
Z ( z )  supminA( x), B( x) z  x  y
x, y
Horizontal representation of this operation is
z1  , z2    a1  , a2    b1  , b2  
Difference of fuzzy numbers
Then membership function for
~ ~ ~
Z = A-B
is defined as
Z ( z )  supminA( x), B( x) z  x  y
x, y
Horizontal representation of this operation is
z1  , z2    a1  , a2    b1  , b2  
Product of fuzzy numbers
Then membership function for
~ ~ ~
Z = A.B
is defined as
Z ( z )  supminA( x), B( x) z  x. y
x, y
Horizontal representation of this operation is
z1  , z2    a1  , a2   . b1  , b2  
Fraction of fuzzy numbers
Then membership function for
~ ~ ~
Z = A/B
is defined as
Z ( z )  supminA( x), B( x) z  x / y
x, y
Horizontal representation of this operation is
z1  , z2    a1 , a2   / b1  , b2  
Function on fuzzy numbers
x )= ~y
A fuzzy function is a mapping from fuzzy numbers into fuzzy numbers. We write h( ~
x . For two independent variables we
for a fuzzy function with one independent variable ~
~
~
~
~
y in two ways
have h( x , y )= z . Let h:a,bR. We extend h( x )= ~
a) the extension principle
b) -cuts and interval arithmetic.
Extension principle
Let h:a,bR and
membership function is
~
x
is fuzzy number(usually triangular or trapezoidal). Then
h( ~
x ) z   ~
y  z   sup~
x  x ; h( x)  z, z  a, b 
x
Example
x is triangular fuzzy number (-1,1,2). What is h( ~
x )?
Let h(x)=x2 and ~
x ) is fuzzy number. Membership function of ~
x is
It is clear h( ~
 0, x   1,2
1
~
x ( x)    x  1; x   1,1
2
 2  x; x  1,2
x )= ~y is
The membership function of h( ~
h( ~
x ) z   ~
y  z   sup ~
x  x ; x 2  z , x   1,2

x
x is -1,2 and
The support of ~
y is 0,4,
the support of ~
~
x (x)-1,0 ~
x (x)0,1 and
~y (z)= 1 z  1 ,z0,1
2
y (z)= 2  z ,z1,4
and ~





-cuts and interval arithmetic
If h is continuous, then we can find -cuts of
Where
~y . Let ~y ()=y1(),y2() .
y1    minh( x) x  ~
x  
y2    maxh( x) x  ~
x  ,0,1
Example
Let h(x)=x2 and
~
x
x )?
is triangular fuzzy number (-1,1,2). What is h( ~
x ) is fuzzy number. Membership function of
It is clear h( ~
 0, x   1,2
1
~
x ( x)    x  1; x   1,1
2
 2  x; x  1,2
~
x
is
~
x are -1+2,2-
y1    minh( x) x  ~
x    min x 2 x   1  2 ,2   
-cuts of


0,  0.5


2



1

2

,0.5    1

y2    max h( x) x  ~
x    max x 2 x   1  2 ,2    2   2 ,

0,1

Metrics on fuzzy numbers
If x,y are real numbers, then their distance is d=x-y an is contents
a of
parallelogram(see fig). We use this geometrical notation to define a distance of two fuzzy
numbers.
Let f:A, BC, D is a map interval A, B onto interval C, D. Then
sup x; f  x   y

x A , B
f 1  y   
 A; f x) y for all x  A, B
is pseudo inverse function to f.

Metrics on fuzzy numbers
~
Let a~ , b are two fuzzy numbers and La  x , Lb  x  are left parts
their membership functions and Pa  x , Pb  x  are right parts their
membership functions. Then
1
d a, b    ( L1a  y   L1b  y   Pa
1
 y   P 1b  y  )dy
0
is distance of fuzzy numbers
~
~
a,b .
Metrics on fuzzy numbers
Infimum and supremum of fuzzy numbers
Let
~
a~ , b
are two fuzzy numbers then
 inf a ,b  z    a b  z   supmin a  x ,  b  y  z  minx, y, x, y  R
 supa ,b  z    a b  z   supmin a  x ,  b  y  z  max x, y, x, y  R
Infimum
Supremum
~
a~  b
~
a~  b
Comparable fuzzy numbers
Let
~
~
a,b
are two fuzzy numbers then fuzzy number
a~ 
~
a~  b
or
a~
is greater or equal
~
~
b  a~  b
~
b if
Slovak University of Technology
Faculty of Material Science and Technology in Trnava
Fuzzy Systems
Fuzzy linear equation
Fuzzy linear equation
~
Let a~, b , c~, ~x are fuzzy numbers then
~
a~~
x  b  c~
is fuzzy linear equation.
Problem: How to solve it?
Major problem in solving fuzzy equations
If a, b, c, xR and a0 then x 
cb
~ ~
~ / a~  1
, but b  b  0 and a
a
~ is triangular fuzzy number (a, b, c) then
Example: Let a
a c
~
~
~
~
a - a =(a-c,0,c-a)(0,0,0) and a / a   ,1,   1. For instance if a~ =(1,2,3)
c a
~ - a~ =(-2,0,2) and a~ / a~   1 ,1,3   1 .
then a
3 
This is a major problem in solving fuzzy equations.
Classical method solving of fuzzy equation
Let
~
a~ , b  , c~ , ~
x  
~, b~, c~, ~
x . Then fuzzy equation we can express as
are -cuts of a
~
~
~
a  x    b    c~ 
~
~
a    a1  , a2   , b    b1  , b2   ,
If
c~    c1  , c2   , ~
x    x1  , x2  
a1  , a2   x1  , x2    b1  , b2    c1  , c2  
This solution (when it exists) we denote as
~
xc
Necessary conditions
If classical solution
~
xc  x1  , x2   exists then
1. x1   is monotonically increasing
2. x2   is monotonically decreasing
3. x1    x2   for all 0 1
Example
Let
~
xc
does not exists
~
a~  1,2,3 , b   3,2,1 , c~  3,4,5  Then its -cuts are
a1  , a2    1   ,3  
,
b1  , b2     3   ,1  
c1  , c2    3   ,5  
and
a1  , a2   x1  , x2    b1  , b2    c1  , c2  
a1  x1    b1  , a2  x2    b2    c1  , c2  
1   x1    3   , 3   x2    1  
x1  
6
6
x1   
, x2   
1
3 
~
x
is decreasing and so
c does not exists.
 3   ,5  
is
~
Example xc exists
Let
~
a~  1,2,3 , b   3,2,1 , c~   3,0,3 Then its -cuts are
a1  , a2    1   ,3  
,
b1  , b2     3   ,1  
c1  , c2     3  3 ,3  3
and
a1  , a2   x1  , x2    b1  , b2    c1  , c2  
a1  x1    b1  , a2  x2    b2    c1  , c2  
1   x1    3   , 3   x2    1  
  3  3 ,3  3
2
2
4  2
2
x1   
 2
, x2   
 2
1
1
3 
3 
x1  
x  
is increasing, 2
is decreasing and
2
and so
~
xc
2
2
 2
 2  6  2  4  6  2
1
3 
exists.
is
Extended principle of solution fuzzy equation
Let
~ ~ ~
~
a,b ,c , x
are fuzzy numbers and
~ ~
~
~
ax  b  c
is fuzzy equation then it too
often has not solution. The fuzzified crisp solution is
evaluate this formula in two ways.
1. extension principle
2. -cuts and interval arithmetic.
If
~
~
x  (c~  b ) / a~ .
~
xe is evaluated by extension principle then its membership function is
~
x  x   max u a, b, c  c  b  / a  x 
e
where


~
u (a, b, c)  min a~a , b b , c~c 
We can
-cuts and interval arithmetic of solution fuzzy
If the result is
~
xI
then
~    b~  
c
~
x I   
a~  
~
xc exists then ~
xc ~
xe
~ x
Theorem: xe ~
I
Theorem: If
Note: For more complicated fuzzy equations
will be difficult to compute. For this
~
~
reason we suggest approximating x by xc and ,
e
I
~
x
An example of solution fuzzy equation
Let
~
~
a  1,2,3 , b   3,2,1 , c~  3,4,5 Then its -cuts are
a1  , a2    1   ,3  
,
b1  , b2     3   ,1  
c1  , c2    3   ,5  
~
~
c    b   3   ,5     3   ,1  
~
xI   


~
a  
1   ,3  
4  2 8  2

,
3  1
Evaluating of fuzzy formulas
Let f: ARR is real function of real variable(s). We usually compute its value apply finite
number of basic arithmetic operations. Fur instance
3
5
7
x
x
x
sin( x)  x  

6 120 5040
~
The image of fuzzy number x is evaluated in two methods
1. extension principle
2. -cuts and interval arithmetic.
~
y
If it is used extension principle then membersip function of fuzzy number
f ~
x is
 
~
y  z   sup~
x  x  f  x   z
x
~
y  f ~
x  is
y   y1 , y2    min f x  x  x , max f x  x  x 
If f is continuous, then -cuts of
Alpha-cuts and interval arithmetic
All the functions we use in engineering have algorithm which use finite number of basic
arithmetical operations. For instance
3
5
7
sin( x)  x 
x
x
x


6 120 5040
In fuzzy mathematic we have the interval
interval arithmetic. For instance
x 
and we perform needed operations
x3   x5   x 7  
sin( x )  x  


6
120
5040
Example
Let f(x)=x(1-x)=y, ~y  f ~x   ~x 1  ~x  and x   x1 , x2   .
Let ~x is triangular fuzzy number (0,0.25,0.5).
Extension principle
~
y  f ~
x  and
is x
    , 1  
4 2 4
y   y1 , y2    min f x  x  x , max f x  x  x 
y   minx(1  x) x  x , maxx(1  x) x  x  =
=
4   2 4   2
,
16
16
-cuts and interval arithmetic
Let f(x)=x(1-x)=y,
~
x
~
y  f ~
x ~
x 1  ~
x  and x   x1 , x2  
is triangular fuzzy number (0,0.25,0.5). Then -cut of
y    x 1  x  

 1 
, 
4 2 4
 1  
~
x
is
x  
. Let
 1 
, 
4 2 4
 1  
,  1 
, 

4 2 4 
4 2 4 
1 

2   2 8  6   2
 ,1 

,
 y*  
2 4
4
16
16
and
Slovak University of Technology
Faculty of Material Science and Technology in Trnava
Fuzzy Systems
Fuzzification and defuzzification
Fuzzification and defuzzification
As a result of applying the previous steps, one obtains
a fuzzy set from the reasoning process that describes, for
each possible value, how reasonable it is to use this
particular value. In other words, for every possible value,
one gets a grade of membership that describes to what
extent this value is reasonable to use. Using a fuzzy system
as a controller, one wants to transform this fuzzy
information into a single value that will actually be applied.
This transformation from a fuzzy set to a crisp number is
called a defuzzification. It is not a unique operation as
different approaches are possible. The most important ones
for control are described in the following.
Center of area or center of gravity method
(COG)
This approach has its origin in the idea to select a value  that, on average, would lead to
the smallest error in the sense of a criterion. If  is chosen, and the best value is x then
the error is x-. Thus, to determine x the least squares method can be used. As weights
for each square (x-)2, one can take the grade of membership  with which x is a
reasonable value. As a result one has to find


2
2
  min   ( x)x  a  dx  min   x  ( x)dx  2a  x ( x)dx  a 
a
a 

u
U
U
2
 2  x ( x)dx  2a  0
U
a   x x dx
U
The center of area or center of gravidity is
 x x dx
 U
  x dx
U
Center of area or center of gravity method.
Discrete fuzzy set
If fuzzy set is discrete then
  min
a




  ( x)x  a 2  min  x 2  ( x)  2a x ( x)  a 2 
U
a

U
U
 x ( x)

U
  x 
U


Example
0; x1

 x  1, x  1,2

Let   x    4  x
. Then defuzzificated value is
; 2,4

2

0, x 4

 x x dx
2
4
4 x


x

1
xdx

xdx


5
29

4
2
29
U
1
2


 6
 6 
  x dx 2 x  1dx  4 4  x xdx 1  1 3 9
2
2

 2
U
1
2
Center of sum (COS)
The defuzzification can be strongly simplified if the membership functions  of the
conclusions are singly defuzzified for each rule such that each function is reduced to a
singleton that has the position i of the individual membership function's centre of gravity.
The centre of singletons is calculated by using the degree of relevance as follows:
  i si

i
 si
i
The simplification consists in that the singletons can be determined already during
the design of the fuzzy system and that the membership function  with its complicated
geometry is no longer needed. The defuzzification using this formula is an approximation
of the defuzzification. Experiences from control show that there are slight differences
between both approaches, which can be in most cases neglected.
First of maximum methods (FoM)
This class of methods determines  by selecting the membership function with
the maximum value. If the maximum is a range, the lower, upper or the middle value is
taken for  depending on the method. Using these methods, the rule with the
maximum activity always determines the value, and therefore they show discontinuous
and step output on continuous input. This is the reason why these types of method are
not attractive for use in controllers.
Last of maximum methods (LoM)
This class of methods determines  by selecting the
membership function with the last maximum value. If the
maximum is a range, the upper or the middle value is taken
for  depending on the method. Using these methods, the
rule with the maximum activity always determines the value,
and therefore they show discontinuous and step output on
continuous input. This is the reason why these types of
method are not attractive for use in controllers.
Margin properties of the centroid methods
As the centre of gravity of the area below the
membership functions cannot reach the margins of x, the
membership functions, which are at the margins, must be
symmetrically expanded when obtaining the centre of
gravity. This is necessary in order to have the full range of x
available.
Margin of  (a) original and (b) expanded
The methods of defuzzification













RCOM (random choice of maximum)
FOM (first of maximum)
LOM (last of maximum)
MOM (middle of maximum)
COG (center of gravity)
MeOM (mean of maxima)
BADD (basic defuzzification distributions)
GLSD (generalized level set defuzzification)
ICOG (indexed center of gravity)
SLIDE (semi-linear defuzzification)
FM (fuzzy mean)
WFM (weighted fuzzy mean)
QM (quality method)
The methods of defuzzification





EQM (extended quality method)
COA (center of area)
ECOA (extended center of area)
CDD (constraint decision defuzzification)
FCD (fuzzy clustering defuzzification)
The maxima methods are good candidates for fuzzy
reasoning systems. The distribution methods and the area
methods exhibit the property of continuity that makes them
suitable for fuzzy controllers .
Defuzzification: criteria and classification, from the
journal Fuzzy Sets and Systems, Van Leekwijck and Kerre,
Vol. 108 (1999), pp. 159-178
Linguistic Variable
Linguistic Variable - Linguistic means relating to language, in our case plain
language words. Speed is a fuzzy variable. Accelerator setting is a fuzzy variable.
Examples of linguistic variables are: somewhat fast speed, very high speed, real slow
speed, excessively high accelerator setting, accelerator setting about right, etc.
A fuzzy variable becomes a linguistic variable when we modify it with descriptive
words, such as somewhat fast, very high, real slow, etc. The main function of linguistic
variables is to provide a means of working with the complex systems mentioned above as
being too complex to handle by conventional mathematics and engineering formulas.
Linguistic variables appear in control systems with feedback loop control and
can be related to each other with conditional, "if-then" statements. Example: If the
speed is too fast, then back off on the high accelerator setting.
Universe of Discourse
Universe of Discourse - Let us make women the object
of our consideration. All the women everywhere would be
the universe of women. If we choose to discourse about
(talk about) women, then all the women everywhere would
be our Universe of Discourse.
Universe of Discourse then, is a way to say all the
objects in the universe of a particular kind, usually
designated by one word, that we happen to be talking about
or working with in a fuzzy logic solution.
A Universe of Discourse is made up of fuzzy sets.
Example: The Universe of Discourse of women is made up
of professional women, tall women, Asian women, short
women, beautiful women, and on and on.
Fuzzy Algorithm
Fuzzy Algorithm - An algorithm is a procedure, such as
the steps in a computer program. A fuzzy algorithm, then,
is a procedure, usually a computer program, made up of
statements relating linguistic variables.
Examples:
If "green x" is very large, then make "tall y" much smaller.
If the rate of change of temperature of the steam
engine boiler is much too high then turn the heater down a
lot.
Zadeh's original definition of a linguistic variable is
rather inspired by computational linguistics and classical AI
and much more sophisticated than the shallow
understanding that is most often used in engineeringoriented domains like fuzzy control.
Linguistic Variable
Usually, a linguistic variable is a quintuple (L, G, T, U, S),
where L, T, U, G, and S are defined as follows:
1. L is the name of the linguistic variable V (label)
2. G is a grammar
3. T is the so-called term set, i.e. the set linguistic expressions resulting from G
4. U is the universe of discourse
5. S is a TF(X) mapping which defines the semantics - a fuzzy set on X -of each
linguistic expression in T.