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PART 3Operations on fuzzy sets
1. Fuzzy complements2. Fuzzy intersections3. Fuzzy unions4. Combinations of operations5. Aggregation operations
FUZZY SETS AND
FUZZY LOGICTheory and Applications
Fuzzy complements
• Axiomatic skeleton
Axiom c1.
Axiom c2.
(boundary conditions).
For all if , then (monotonicity).
01 and 10 cc
]1,0[, ba ba )()( bcac
Fuzzy complements
• Desirable requirements
Axiom c3.
Axiom c4.
c is a continuous function.
c is involutive, which means that for each
aacc ))((.]1,0[a
Fuzzy complements
• Theorem 3.1Let a function satisfy Axioms c2 and c4. Then, c also satisfies Axioms c1 and c3. Moreover, c must be a bijective function.
]1 ,0[]1 ,0[: c
Fuzzy complements
Fuzzy complements
Fuzzy complements
• Sugeno class
• Yager class
). ,1( where,1
1)(
a
aac
). ,0( where,)1()( /1 waac www
Fuzzy complements
Fuzzy complements
• Theorem 3.2Every fuzzy complement has at most one equilibrium.
Fuzzy complements
• Theorem 3.3
. iff
iff
c
c
eaaca
eaaca
Assume that a given fuzzy complement c has an equilibrium ec , which by Theorem 3.2 is unique. Then and
Fuzzy complements
• Theorem 3.4
• Theorem 3.5
If c is a continuous fuzzy complement, then c has a unique equilibrium.
If a complement c has an equilibrium ec , then
. ccd ee
Fuzzy complements
Fuzzy complements
• Theorem 3.6For each , , that is, when the complement is involutive.
]1 ,0[a aaccacad ))(( iff )(
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 from [0, 1] to R such that , is strictly increasing, and
for all
0)0( g
))()1(()( 1 agggac
].1 ,0[a
g g
Fuzzy complements
• Increasing generators
Sugeno:
Yager:
.1for 1ln1
aag
.0for waag ww
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 from [0, 1] to R such that , is strictly decreasing, and
for all .
01 ff
afffac 01
]1 ,0[a
f
Fuzzy complements
• Decreasing generators
Sugeno:
Yager: .0 where,1)( waaf w
.1 where
),1ln(1
)1ln()(
aaaf
Fuzzy intersections: t-norms
• Axiomatic skeleton
Axiom i1.
Axiom i2.
(boundary condition).
implies (monotonicity).
aai 1,
db daibai ,,
Fuzzy intersections: t-norms
• Axiomatic skeleton
Axiom i3.
Axiom i4.
(commutativity).
(associativity).
abibai ,,
dbaiidbiai ,,,,
Fuzzy intersections: t-norms
• Desirable requirements
Axiom i5
Axiom i6
Axiom i7
is a continuous function (continuity).
(subidempotency).
implies
(strict monotonicity).
i
aaai ,
2121 and bbaa
),(),( 2211 baibai
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
otherwise. 0
1 when
1 when
) ,( :onintersecti Drastic
)1 ,0(maxi :difference Bounded
.) ,( :product Algebraic
). ,min() ,( :onintersecti Standard
ab
ba
bai
baa, b
abbai
babai
Fuzzy intersections: t-norms
Fuzzy intersections: t-norms
Fuzzy intersections: t-norms
• Theorem 3.9
• Theorem 3.10
The standard fuzzy intersection is the only idempotent t-norm.
For all ,
where denotes the drastic intersection.
]1 ,0[, ba
, ,min , ,min babaibai
mini
Fuzzy intersections: t-norms
• Pseudo-inverse of decreasing generator
The pseudo-inverse of a decreasing generator , denoted by , is a function from R to [0, 1] given by
where is the ordinary inverse of .
f )1(f
f) ),0((for
)]0( ,0[for
)0 ,(for
0
)(
1
)( 1)1(
fa
fa
a
afaf
)1(f
Fuzzy intersections: t-norms
• Pseudo-inverse of increasing generator
The pseudo-inverse of a increasing generator , denoted by , is a function from R to [0, 1] given by
where is the ordinary inverse of .
g )1(g
g) ),1((for
)]1( ,0[for
)0 ,(for
1
)(
0
)( 1)1(
ga
ga
a
agag
)1(g
Fuzzy intersections: t-norms
• Lemma 3.1
Let be a decreasing generator. Then a function defined by
for any is an increasing generator
with , and its pseudo-inverse
is given by
for any R.
fg
)()0()( affag ]1 ,0[a)0()1( fg )1(g
a))0(()( )1()1( affag
Fuzzy intersections: t-norms
• Lemma 3.2
Let be a increasing generator. Then a function defined by
for any is an decreasing generator
with , and its pseudo-inverse
is given by
for any R.
gf
)()1()( aggaf ]1 ,0[a)1()0( gf )1(f
a))1(()( )1()1( aggaf
Fuzzy intersections: t-norms
• Theorem 3.11 (Characterization Theorem of t-Norms).
Let be a binary operation on the unit interval. Then, is an Archimedean t-norm iff there exists a decreasing generator
such that
for all .
f
]1 ,0[ , ba
))()(() ,( )1( bfaffbai
ii
Fuzzy intersections: t-norms
• [Schweizer and Sklar, 1963]
.))1 ,0(max(
otherwise.
]1 ,0[2when
0
)1(
)2(
))()(() ,(
) ,1( where
]1 ,0[ where
)0 ,( where
0
)1(
1
)(
).0( 1)(
1
1
)1(
)1(
p1)1(
pPp
ppppp
ppp
pppp
p
pp
ba
baba
baf
bfaffbai
z
z
z
zzf
paaf
Fuzzy intersections: t-norms
• [Yager, 1980f]
).])1()1([ ,1min(1
otherwise.]1 ,0[)1()1(
when
0
))1()1((1
))1()1((
))()(() ,(
) ,1( where
]1 ,0[ where
0
1)(
),0( )1()(
1
1
)1(
)1(
1)1(
www
ww
www
www
wwww
w
w
ww
ba
baba
baf
bfaffbai
z
zzzf
waaf
Fuzzy intersections: t-norms
• [Frank, 1979]
.1
)1)(1(1log
)1(
)1)(1()1(1log
)1(
)1)(1(ln
))()(( ) ,(
).)1(1(log)(
),1 ,0( 1
1ln)(
2
2)1(
)1(
)1(
s
ss
s
sss
s
ssf
bfaffbai
eszf
sss
saf
ba
s
ba
s
ba
s
ssss
zss
a
s
Fuzzy intersections: t-norms
• Theorem 3.12
Let denote the class of Yager t-norms.
Then,
for all , where the lower and upper bounds are obtained for and
,respectively.
wi
) ,min() ,() ,(min babaibai w
]1 ,0[ , ba0w w
Fuzzy intersections: t-norms
• Theorem 3.13
Let be a t-norm and be a
function such that is strictly increasing
and continuous in (0, 1) and
Then, the function defined by
for all ,where denotes the pseudo-inverse of , is also a t-norm.
i ]1 ,0[]1 ,0[: gg
.1)1( ,0)0( gggi
)))( ),((() ,( )1( bgagigbai g
]1 ,0[ , bag
)1(g
Fuzzy unions: t-conorms
• Axiomatic skeleton
Axiom u1.
Axiom u2.
).( )0 ,( onditionboundary caau
).( ) ,() ,( implies tymonotonicidaubaudb
Fuzzy unions: t-conorms
• Axiomatic skeleton
Axiom u3.
Axiom u4.
).( ) ,() ,( itycommutativabubau
).( ) ), ,(()) ,( ,( ityassociativdbauudbuau
Fuzzy unions: t-conorms
• Desirable requirements
Axiom u5.
Axiom u6.
Axiom u7.
).(function continuous a is continuityu
).(s ) ,( tencyuperidempoaaau
).( )()(
implies and
2211
2121
otonicitystrict mon, bau, bau
bbaa
Fuzzy unions: t-conorms
• Frequently used t-conorms
otherwise.
0when
0when
1
) ,( :union Drastic
). ,1min() ,( :sum Bounded
.) ,( :sum Algebraic
). ,max() ,( :union Standard
]1 ,0[ , allfor
a
b
b
a
bau
babau
abbabau
babau
ba
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
],1 ,0[ , ba
). ,() ,() ,max( max baubauba
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 t-conorm iff there exists an increasing generator such that
for all ].1 ,0[ , ba
))()(() ,( )1( bgaggbau
Fuzzy unions: t-conorms
• [Schweizer and Sklar, 1963]
.)1)1()1( ,0max(1
otherwise.
]1 ,0[)1()1(2when
1
]1)1()1[(1
))1(1)1(1() ,(
) ,1(when
]1 ,0[when
1
)1(1)(
).0( ) 1(1)(
1
1
)1(
1)1(
ppp
ppppp
ppp
p
p
p
pp
ba
baba
bagbau
z
zzzg
paag
Fuzzy unions: t-conorms
• [Yager, 1980f]
).)( ,1min(
)() ,(
) ,1(when
]1 ,0[when
1)(
),0( )(
1
)1(
1)1(
www
wwww
w
w
ww
ba
bagbau
z
zzzg
waag
Fuzzy unions: t-conorms
• [Frank, 1979]
.1
)1)(1(1log1) ,(
),)1(1(log1)(
)1 ,0( 1
1ln)(
11
)1(
1
s
ssbau
eszg
sss
sag
ba
ss
zss
a
s
Fuzzy unions: t-conorms
• Theorem 3.17
Let uw denote the class of Yager t-conorms.
for all where the lower and upper bounds are obtained for ,
respectively.
) ,() ,() ,max( max baubauba w
]1 ,0[ , ba
0 and ww
Fuzzy unions: t-conorms
• Theorem 3.18
Let u be a t-conorm and let be
a function such that is strictly increaning
and continuous in (0, 1) and .
Then, the function defined by
for all is also a t-conorm.
]1 ,0[]1 ,0[: g
g
1)1( ,0)0( gggu
]1 ,0[ , ba
)))( ),((() ,( )1( bgagugbau g
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
for all is a t-conorm such that
〈 i, u, c 〉 is a dual triple.
]1 ,0[ , ba
)))( ),((() ,( bcacicbau
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
for all is a t-norm such that
〈 i, u, c 〉 is a dual triple.
]1 ,0[ , ba
)))( ),((() ,( bcacucbai
Combinations of operators
• Theorem 3.22
Given an involutive fuzzy complement c and an increasing generator of c, the
t-norm and t-conorm generated by are dual with respect to c.
g
g
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.
Axiom h2.
). ( 1)1 ..., ,1 ,1( and 0)0 ..., ,0 ,0( onditionsboundary chh
arguments. its allin is is,that
; )()(
then
, allfor if , allfor ]10[such that
tuples- of ,... , , and ,... , ,pair any For
2121
2121
ingincreasmonotonic h
, ..., b, bbh, ..., a, aah
NibaNi, , ba
nbbbaaa
nn
niinii
nn
Aggregation operations
• Axiomatic requirements
Axiom h3.
function. is continuoush
Aggregation operations
• Additional requirements
Axiom h4.
Axiom h5.
.on n permutatioany for
)()(
is, that arguments; its allin function a is
)()2()1(21
n
npppn
Np
, ..., a, aah, ..., a, aah
symmetrich
].1 ,0[ allfor
)(
is, that function; an is
a
a aa, a, ...,h
idempotenth
Aggregation operations
• Theorem 3.25
.
121
21212211
n
N allfor 0 where
,)(
Then, .N allfor 1] [0, where
)()()(
property theand h2,
Axiomh1, Axiom satisfieshat function t a be R1] [0,:Let
ni
n
iiin
niiii
nnnn
iw
aw, ..., a, aah
ib, a, ba
, ..., b, bbh, ..., a, aahb, ..., ab, abah
h
Aggregation operations
• Theorem 3.26
.
1121
212111
N allfor 1] [0, where
),) ,min( ..., ), ,max(min()(
Then, .N allfor 0) ..., 0, , ...,0, (0,)( where
)())( (
))( ),(max()) ,max() ,(max(
property theand h3,
Axiomh1, Axiom satisfieshat function t a be 1] [0,1] [0,:Let
ni
nnn
niii
iiiii
nnnn
n
iw
awaw, ..., a, aah
iahah
ahahh
, ..., b, bbh, ..., a, aahba, ..., bah
h
Aggregation operations
• Theorem 3.27
).min()(
such that 1] [0,..., numbers
exist thereThen, .N allfor 1) ..., 1, , ...,1, (1,)( where
0)0( and )()( (ab)
))( ),(min()) ,min() ,(min(
property theand h3,
Axiomh1, Axiom satisfieshat function t a be 1] [0,1] [0,:Let
21
2121
21
212111
nαn
ααn
n
niii
iiii
nnnn
n
, ..., a, aa, ..., a, aah
α,α,α
iahah
hbhahh
, ..., b, bbh, ..., a, aahba, ..., bah
h
Aggregation operations
• Theorem 3.28
1]. [0,any for
otherwise
]1 ,[ where
] [0, where
)min(
) ,max(
) ,(
such that 1] [0, exists thereThen,
.idempotent and continuous be operation norm aLet
a, b
a, b
a, b
a, b
ba
bah
h
Exercise 3
• 3.6
• 3.7
• 3.13
• 3.14
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