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FUZZY SETS
AND
FUZZY LOGIC
Theory and
Applications
PART 3
Operations on
fuzzy sets
1. Fuzzy complements
2. Fuzzy intersections
3. Fuzzy unions
4. Combinations of operations
5. Aggregation operations
Fuzzy complements
• Axiomatic skeleton
Axiom c1.
c0  1 and c1  0 (boundary conditions).
Axiom c2.
For all a, b  [0,1] if a  b , then c(a)  c(b) (monotonicity).
Fuzzy complements
• Desirable requirements
Axiom c3.
c is a continuous function.
Axiom c4.
c is involutive, which means that c(c(a))  a for each
a  [0,1].
Fuzzy complements
• Theorem 3.1
Let a function c : [0, 1]  [0, 1] satisfy Axioms c2 and c4.
Then, c also satisfies Axioms c1 and c3. Moreover, c
must be a bijective function.
Fuzzy complements
Fuzzy complements
Fuzzy complements
• Sugeno class
1 a
c (a ) 
, where   (1, ).
1  a
• Yager class
cw (a)  (1  a ) , where w  (0, ).
w 1/ w
Fuzzy complements
Fuzzy complements
• Theorem 3.2
Every fuzzy complement has at most one equilibrium.
Fuzzy complements
• Theorem 3.3
Assume that a given fuzzy complement c has an
equilibrium ec , which by Theorem 3.2 is unique. Then
a  ca  iff a  ec
and
a  ca  iff a  ec .
Fuzzy complements
• Theorem 3.4
If c is a continuous fuzzy complement, then c has a
unique equilibrium.
• Theorem 3.5
If a complement c has an equilibrium ec , then
d
ec  ec .
Fuzzy complements
Fuzzy complements
• Theorem 3.6
For each a  [0, 1] , a  c(a) iff c(c(a))  a , that is,
when the complement is involutive.
d
Fuzzy complements
• Theorem 3.7
(First Characterization Theorem of Fuzzy Complements).
Let c be a function from [0, 1] to [0, 1]. Then, c is a fuzzy
complement (involutive) iff there exists a continuous
function g from [0, 1] to R such that g (0)  0 , g is
strictly increasing, and
c(a)  g 1 ( g (1)  g (a))
for all a  [0, 1].
Fuzzy complements
• Increasing generators
Sugeno:
g  a  
1

ln1  a  for   1.
Yager:
gw a  a w for w  0.
Fuzzy complements
• Theorem 3.8
(Second Characterization Theorem of Fuzzy complements).
Let c be a function from [0, 1] to [0, 1]. Then c is a fuzzy
complement iff there exists a continuous function f from
[0, 1] to R such that f 1  0 , f is strictly decreasing, and
ca   f 1  f 0  f a 
for all a  [0, 1] .
Fuzzy complements
• Decreasing generators
Sugeno:
f (a)  ln(1  a) 
where   1.
1

ln(1  a),
Yager:
f (a)  1  a w , where w  0.
Fuzzy intersections: t-norms
• Axiomatic skeleton
Axiom i1.
ia,1  a (boundary condition).
Axiom i2.
b  d implies ia, b   ia, d  (monotonicity).
Fuzzy intersections: t-norms
• Axiomatic skeleton
Axiom i3.
ia, b  ib, a  (commutativity).
Axiom i4.
ia, ib, d   iia, b, d 
(associativity).
Fuzzy intersections: t-norms
• Desirable requirements
Axiom i5
i
is a continuous function (continuity).
Axiom i6
ia, a   a
(subidempotency).
Axiom i7
a1  a2 and b1  b2 implies
i(a1 , b1 )  i(a2 , b2 ) (strict monotonicity).
Fuzzy intersections: t-norms
• Archimedean t-norm:
A t-norm satisfies Axiom i5 and i6.
• Strict Archimedean t-norm:
Archimedean t-norm and satisfies Axiom
i7.
Fuzzy intersections: t-norms
• Frequently used t-norms
Standard intersection : i (a, b)  min(a, b).
Algebraic product : i (a, b)  ab.
Bounded difference: ia, b  max(0, a  b  1)
a when b  1

Drasticintersection : i (a, b)  b when a  1
 0 otherwise.

Fuzzy intersections: t-norms
Fuzzy intersections: t-norms
Fuzzy intersections: t-norms
• Theorem 3.9
The standard fuzzy intersection is the only idempotent
t-norm.
• Theorem 3.10
For all a, b  [0, 1] ,
imin a, b  ia, b  mina, b,
where imin denotes the drastic intersection.
Fuzzy intersections: t-norms
• Pseudo-inverse of decreasing generator
The pseudo-inverse of a decreasing
(1)
generator f , denoted by f , is a function
from R to [0, 1] given by
for a  (, 0)
1
 1
( 1)
f (a)   f (a) for a  [0, f (0)]
0
for a  ( f (0), )

(1)
f
where
is the ordinary inverse of f .
Fuzzy intersections: t-norms
• Pseudo-inverse of increasing generator
The pseudo-inverse of a increasing
(1)
g
generator , denoted by g , is a function
from R to [0, 1] given by
for a  (, 0)
0
 1
( 1)
g (a )   g (a ) for a  [0, g (1)]
1
for a  ( g (1), )

(1)
g
where
is the ordinary inverse of g.
Fuzzy intersections: t-norms
• Lemma 3.1
Let f be a decreasing generator. Then a
function g defined by
g (a)  f (0)  f (a)
for any a  [0, 1] is an increasing generator
with g (1)  f (0), and its pseudo-inverse g (1)
is given by
g ( 1) (a)  f ( 1) ( f (0)  a)
for any a R.
Fuzzy intersections: t-norms
• Lemma 3.2
Let g be a increasing generator. Then a
function f defined by
f (a)  g (1)  g (a)
for any a  [0, 1] is an decreasing generator
with f (0)  g (1), and its pseudo-inverse f (1)
is given by
f ( 1) (a)  g ( 1) ( g (1)  a)
for any a R.
Fuzzy intersections: t-norms
• Theorem 3.11 (Characterization Theorem of t-Norms).
Let i be a binary operation on the unit
interval. Then, i is an Archimedean t-norm
iff there exists a decreasing generator f
such that
i(a, b)  f
for all a, b  [0, 1].
( 1)
( f (a)  f (b))
Fuzzy intersections: t-norms
• [Schweizer and Sklar, 1963]
f p (a)  1  a p ( p  0).
where z  (, 0)
1

( 1)
f p ( z )  (1  z )1 p where z  [0, 1]
0
where z  (1, )

i p (a, b)  f p( 1) ( f p (a)  f p (b))
 f p( 1) (2  a p  b p )
(a p  b p  1)1 p when 2  a p  b p  [0, 1]

otherwise.
0
 (max(0, a p  b P  1))1 p .
Fuzzy intersections: t-norms
• [Yager, 1980f]
f w (a)  (1  a) w ( w  0),
1w
where z  [0, 1]

1

z
( 1)
f w ( z)  
where z  (1, )
0
iw (a, b)  f w( 1) ( f w (a )  f w (b))
 f w( 1) ((1  a) w  (1  b) w )
1  ((1  a)  (1  b) )

0
w
w 1w
when
(1  a ) w  (1  b) w  [0, 1]
otherwise.
 1  min(1, [(1  a) w  (1  b) w ]1 w ).
Fuzzy intersections: t-norms
• [Frank, 1979]
sa 1
f s (a)   ln
( s  0, s  1),
s 1
( 1)
f s ( z )  logs (1  ( s  1)e  z ).
is (a, b)  f s( 1) ( f s (a)  f s (b))
 f
( 1)
s

( s a  1)(s b  1) 
 ln

2
(
s

1
)



( s a  1)(s b  1) 
 logs 1  ( s  1)

2
(
s

1
)


 ( s a  1)(s b  1) 
 logs 1 
.
s 1


Fuzzy intersections: t-norms
• Theorem 3.12
Let iw denote the class of Yager t-norms.
Then,
imin (a, b)  iw (a, b)  min(a, b)
for all a, b  [0, 1], where the lower and upper
bounds are obtained for w  0 and w  
,respectively.
Fuzzy intersections: t-norms
• Theorem 3.13
Let i be a t-norm and g : [0, 1]  [0, 1] be a
function such that g is strictly increasing
and continuous in (0, 1) and g (0)  0, g (1)  1.
Then, the function i g defined by
i (a, b)  g
g
( 1)
(i( g (a), g (b)))
for all a, b  [0, 1] ,where g (1) denotes the
pseudo-inverse of g , is also a t-norm.
Fuzzy unions: t-conorms
• Axiomatic skeleton
Axiom u1.
u(a, 0)  a (boundary condition).
Axiom u2.
b  d impliesu(a, b)  u(a, d ) (monotonici
ty).
Fuzzy unions: t-conorms
• Axiomatic skeleton
Axiom u3.
u(a, b)  u(b, a) (commutativ
ity).
Axiom u4.
u(a, u(b, d ))  u(u(a, b), d ) (associativity).
Fuzzy unions: t-conorms
• Desirable requirements
Axiom u5.
u is a continuousfunction(continuity).
Axiom u6.
u(a, a)  a (superidempotency).
Axiom u7.
a1  a2 and b1  b2 implies
u(a1, b1 )  u(a2 , b2 ) (strict monotonicity).
Fuzzy unions: t-conorms
• Frequently used t-conorms
for all a, b  [0, 1]
Standard union : u (a, b)  max(a, b).
Algebraic sum : u (a, b)  a  b  ab.
Bounded sum : u (a, b)  min(1, a  b).
a when b  0

Drasticunion : u (a, b)  b when a  0
1 otherwise.

Fuzzy unions: t-conorms
Fuzzy unions: t-conorms
Fuzzy unions: t-conorms
• Theorem 3.14
The standard fuzzy union is the only
idempotent t-conorm.
Fuzzy unions: t-conorms
• Theorem 3.15
For all a, b [0, 1],
max(a, b)  u(a, b)  umax (a, b).
Fuzzy unions: t-conorms
• Theorem 3.16 (Characterization Theorem
of t-Conorms).
Let u be a binary operation on the unit
interval. Then, u is an Archimedean tconorm iff there exists an increasing
generator such that
u(a, b)  g
for all a, b [0, 1].
( 1)
( g (a)  g (b))
Fuzzy unions: t-conorms
• [Schweizer and Sklar, 1963]
g p (a )  1  (1  a ) p ( p  0).
1 p
when z  [0, 1]

1

(
1

z
)
( 1)
g p ( z)  
when z  (1, )
1
u p (a, b)  g (p1) (1  (1  a ) p  1  (1  b) p )
1  [(1  a ) p  (1  b) p  1]1 p when 2  (1  a ) p  (1  b) p  [0, 1]

otherwise.
1


 1  max(0, (1  a) p  (1  b) p  1)
1 p
.
Fuzzy unions: t-conorms
• [Yager, 1980f]
g w (a )  a w ( w  0),
1w
when z  [0, 1]

z
( 1)
g w ( z)  
1 when z  (1, )
u w (a, b)  g w( 1) (a w  b w )
 min(1, (a w  b w )1 w ).
Fuzzy unions: t-conorms
• [Frank, 1979]
s1 a  1
g s (a )   ln
( s  0, s  1)
s 1
g s( 1) ( z )  1  logs (1  ( s  1)e  z ),
 ( s1 a  1)(s1b  1) 
u s (a, b)  1  logs 1 
.
s 1


Fuzzy unions: t-conorms
• Theorem 3.17
Let uw denote the class of Yager t-conorms.
max(a, b)  uw (a, b)  umax (a, b)
for all a, b  [0, 1] where the lower and upper
bounds are obtained for w   and w  0 ,
respectively.
Fuzzy unions: t-conorms
• Theorem 3.18
Let u be a t-conorm and let g : [0, 1]  [0, 1] be
a function such that g is strictly increaning
and continuous in (0, 1) and g (0)  0, g (1)  1.
g
u
Then, the function defined by
u (a, b)  g
g
( 1)
(u( g (a), g (b)))
for all a, b  [0, 1] is also a t-conorm.
Combinations of operators
• Theorem 3.19
The triples
〈min, max, c〉and〈imin, umax, c〉are dual
with respect to any fuzzy complement c.
Combinations of operators
• Theorem 3.20
Given a t-norm i and an involutive fuzzy
complement c, the binary operation u on
[0, 1] defined by
u(a, b)  c(i(c(a), c(b)))
for all a, b  [0, 1] is a t-conorm such that
〈i, u, c〉is a dual triple.
Combinations of operators
• Theorem 3.21
Given a t-conorm u and an involutive fuzzy
complement c, the binary operation i on
[0, 1] defined by
i(a, b)  c(u(c(a), c(b)))
for all a, b  [0, 1] is a t-norm such that
〈i, u, c〉is a dual triple.
Combinations of operators
• Theorem 3.22
Given an involutive fuzzy complement c
and an increasing generator g of c, the
t-norm and t-conorm generated by g are
dual with respect to c.
Combinations of operators
• Theorem 3.23
Let〈i, u, c〉be a dual triple generated by
Theorem 3.22. Then, the fuzzy operations
i, u, c satisfy the law of excluded middle
and the law of contradiction.
Combinations of operators
• Theorem 3.24
Let〈i, u, c〉be a dual triple that satisfies
the law of excluded middle and the law of
contradiction. Then,〈i, u, c〉does not
satisfy the distributive laws.
Aggregation operations
• Axiomatic requirements
Axiom h1.
h(0, 0, ..., 0)  0 and h(1, 1, ...,1)  1 (boundary onditions
c
).
Axiom h2.
For any pair a1 , a2 , ..., an and b1 , b2 , ..., bn of n - tuples
such thatai , bi  [0 , 1] for all i  N n , if ai  bi for all i  N n ,
then
h(a1, a2 , ..., an )  h(b1, b2 , ..., bn ) ;
thatis, h is m onotonicincreasing in all its arguments.
Aggregation operations
• Axiomatic requirements
Axiom h3.
h is continuous function.
Aggregation operations
• Additional requirements
Axiom h4.
h is a sym m etric functionin all its arguments;thatis,
h(a1, a2 , ..., an )  h(a p (1) , ap ( 2) , ..., ap ( n ) )
for any permutation p on N n .
Axiom h5.
h is an idem potentfunction;thatis,
h(a, a, ...,a)  a
for all a  [0, 1].
Aggregation operations
• Theorem 3.25
Let h : [0,1]n  R  be a function hat
t satisfiesAxiomh1,Axiom
h2, and theproperty
h(a1  b1, a2  b2 , ..., an  bn )  h(a1, a2 , ..., an )  h(b1, b2 , ..., bn )
where ai , bi , ai  bi  [0,1] for all i  N n . T hen,
n
h(a1, a2 , ..., an )   wi ai ,
i 1
where wi  0 for all i  N n.
Aggregation operations
• Theorem 3.26
Let h : [0,1]n  [0,1] be a function hat
t satisfiesAxiomh1,Axiom
h3, and theproperty
h(max(a1 , b1 ) , ..., max(an , bn ))  max(h(a1, a2 , ..., an ), h(b1, b2 , ..., bn ))
hi ( hi (ai ))  hi (ai )
where hi (ai )  h(0,...,0,ai , 0, ..., 0) for all i  N n . T hen,
h(a1, a2 , ..., an )  max(min(w1 , a1 ), ..., min(wn , an )),
where wi  [0,1] for all i  N n .
Aggregation operations
• Theorem 3.27
Let h : [0,1]n  [0,1] be a function hat
t satisfiesAxiomh1,Axiom
h3, and theproperty
h(min(a1 , b1 ) , ..., min(an , bn ))  min(h(a1, a2 , ..., an ), h(b1, b2 , ..., bn ))
hi (ab)  hi (a)hi (b) and hi (0)  0
where hi (ai )  h(1,...,1,ai , 1, ...,1) for all i  N n . T hen,thereexist
numbers α1, α2 , ...,αn  [0,1] such that
α
α
α
h(a1, a2 , ..., an )  min(a1 1 , a2 2 , ..., an n ).
Aggregation operations
• Theorem 3.28
Let a normoperation h be continuousand idempotent.
T hen,thereexists  [0,1] such that
max(a, b) where a, b [0,  ]

h(a, b)  min(a, b) where a, b [ , 1]

otherwise

for any a, b [0,1].
Exercise 3
•
•
•
•
3.6
3.7
3.13
3.14

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