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