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PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations 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

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Page 1: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

PART 3Operations on fuzzy sets

1. Fuzzy complements2. Fuzzy intersections3. Fuzzy unions4. Combinations of operations5. Aggregation operations

FUZZY SETS AND

FUZZY LOGICTheory and Applications

Page 2: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

Fuzzy complements

• Axiomatic skeleton

Axiom c1.

Axiom c2.

(boundary conditions).

For all if , then (monotonicity).

01 and 10 cc

]1,0[, ba ba )()( bcac

Page 3: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

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

Page 4: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

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

Page 5: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

Fuzzy complements

Page 6: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

Fuzzy complements

Page 7: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

Fuzzy complements

• Sugeno class

• Yager class

). ,1( where,1

1)(

a

aac

). ,0( where,)1()( /1 waac www

Page 8: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

Fuzzy complements

Page 9: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

Fuzzy complements

• Theorem 3.2Every fuzzy complement has at most one equilibrium.

Page 10: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

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

Page 11: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

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

Page 12: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

Fuzzy complements

Page 13: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

Fuzzy complements

• Theorem 3.6For each , , that is, when the complement is involutive.

]1 ,0[a aaccacad ))(( iff )(

Page 14: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

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

Page 15: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

Fuzzy complements

• Increasing generators

Sugeno:

Yager:

.1for 1ln1

aag

.0for waag ww

Page 16: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

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

Page 17: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

Fuzzy complements

• Decreasing generators

Sugeno:

Yager: .0 where,1)( waaf w

.1 where

),1ln(1

)1ln()(

aaaf

Page 18: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

Fuzzy intersections: t-norms

• Axiomatic skeleton

Axiom i1.

Axiom i2.

(boundary condition).

implies (monotonicity).

aai 1,

db daibai ,,

Page 19: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

Fuzzy intersections: t-norms

• Axiomatic skeleton

Axiom i3.

Axiom i4.

(commutativity).

(associativity).

abibai ,,

dbaiidbiai ,,,,

Page 20: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

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

Page 21: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

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.

Page 22: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

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

Page 23: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

Fuzzy intersections: t-norms

Page 24: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

Fuzzy intersections: t-norms

Page 25: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

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

Page 26: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

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

Page 27: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

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

Page 28: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

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

Page 29: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

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

Page 30: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

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

Page 31: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

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

Page 32: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

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

Page 33: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

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

Page 34: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

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

Page 35: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

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

Page 36: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

Fuzzy unions: t-conorms

• Axiomatic skeleton

Axiom u1.

Axiom u2.

).( )0 ,( onditionboundary caau

).( ) ,() ,( implies tymonotonicidaubaudb

Page 37: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

Fuzzy unions: t-conorms

• Axiomatic skeleton

Axiom u3.

Axiom u4.

).( ) ,() ,( itycommutativabubau

).( ) ), ,(()) ,( ,( ityassociativdbauudbuau

Page 38: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

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

Page 39: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

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

Page 40: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

Fuzzy unions: t-conorms

Page 41: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

Fuzzy unions: t-conorms

Page 42: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

Fuzzy unions: t-conorms

• Theorem 3.14

The standard fuzzy union is the only idempotent t-conorm.

Page 43: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

Fuzzy unions: t-conorms

• Theorem 3.15

For all

],1 ,0[ , ba

). ,() ,() ,max( max baubauba

Page 44: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

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

Page 45: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

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

Page 46: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

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

Page 47: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

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

Page 48: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

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

Page 49: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

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

Page 50: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

Combinations of operators

• Theorem 3.19

The triples

〈 min, max, c 〉 and 〈 imin, umax, c 〉 are dual

with respect to any fuzzy complement c.

Page 51: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

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

Page 52: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

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

Page 53: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

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

Page 54: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

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.

Page 55: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

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.

Page 56: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

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

Page 57: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

Aggregation operations

• Axiomatic requirements

Axiom h3.

function. is continuoush

Page 58: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

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

Page 59: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

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

Page 60: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

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

Page 61: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

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

Page 62: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

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

Page 63: PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations FUZZY

Exercise 3

• 3.6

• 3.7

• 3.13

• 3.14