CLASSICAL AND FUZZY SETS-VTU.ppt

7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt

1/96

CLASSICAL AND FUZZY SETS

Dr S.Natarajan

Professor,

Department of Information Science and Engineering

PESIT, Bangalore


2/96

2

Classical Sets and Fu! Sets

Classical Sets " #perations on Classical Sets, Properties ofClassical $Crisp% Sets, &apping of Classical Sets to Functions,

Fu! Sets " Fu! Set operations, Properties of Fu! Sets,


3/96

'

( )oolean logic

( fu! logic


4/96

*

Crisp set vs. Fuzzy set

A traditional crisp set A fuzzy set


5/96

5

DEFINITIONS- CLASSICAL SETS

Sets Classical sets either an element belongs to the set or

it does not. For example, for the set of integers, eitheran integer is even or it is not (it is odd). However,either o! are in the "S# or o! are not. $hat abo!t%ing into "S#, what happens as o! are crossing&

#nother example is for blac' and white photographs,one cannot sa either a pixel is white or it is blac'.However, when o! digitie a bw *g!re, o! t!rn allthe bw and gra scales into +5 discrete tones.


6/96

+

Crisp set vs. Fuzzy set


7/96-

# classical(crisp) set Ain the !niverse of disco!rse

Ucan be de*ned in three was/ b enumerating(listing) elements (often called listlistor extensionalextensionalde*nition)

/ b specifing the common propertiesof elements(intensionalintensionalor r!ler!le de*nition)

the notation A = {x !"x#$means that set Aiscomposed of elements x s!ch that everyxhas theproperty!"x#

/ b introd!cing a %ero-one mem&ers'ip (unction

(characteristiccharacteristicorindicatorindicatorde*nition)

3bydivisiblenotis!ndU"i#$"

3bydivisibleis!ndU"i#"%&'('&

3)bydivisibleis*U+ Anu(bers'"inte,ero#&setU

AA


8/960

CLASSICAL SETS ")#

Classical sets are also called crisp(sets).

1ists # 2 3apples, oranges, cherries,mangoes4

# 2 3a,a+,a64

# 2 3+, 7, , 0, 84

Form!las # 2 3x 9 x is an even nat!ral

n!mber4 # 2 3x 9 x 2 +n, n is a nat!ral

n!mber4

:embership or characteristic f!nction

=

Ax

Axx

A if0

if1)(


9/96

9

Characteristic function

Let A be any subset of X, the characteristic

function of A, denoted by , is defined by

Characteristic function of the set of real numbers

from to 10


10/96

-

CLASSICAL SETS &-'/ 0 collection of o)jects all a/ing te same caracteristics 00

1NIE3SE #F DISC#13SE 000 indi/idual elements in te

1ni/erse 4 5ill )e denoted as 6E6amples7

( Te cloc8 speeds of computer CP1s 9ert

( Te operating currents of an electronic motor

( Te operating temperature of a eat pump $ in degrees Celsius%

( Te 3icter magnitude of an eart:ua8e( Te integers to -

First * items are e6amples of real 5orld engineering elements

For te purpose of modeling , tese engineering pro)lems are

simplified and onl! integer /alues of te elements are considered&agnitudes of 3icter Scale greater tan ;


11/96

CLASSICAL SETS &3'

( 1seful metric " cardinalit! or te cardinal num)er " total num)er of

elements in a uni/erse 4 is called its cardinal num)er denoted )! n6,

6 is te inde6

Discrete uni/erses tat are composed of a counta)l! finite collection

of elements 5ill a/e a finite cardinal num)er A continuous uni/erses

comprising of an infinite collection of elements 5ill a/e an infinitecardinalit!

Sets " collections of elements 5itin an uni/erse

Su)sets " collection of elements 5itin te sets


12/96

1!

"asic concepts

#et$ a collection of items

%o &epresent sets

' List method Aa, b, c*' &ule method C x + (x) *

' -amily of sets Ai+ i. *

' /niersal set X and empty set

%he set C is composed of elementsx

ery x has property

i$ inde2 .$ inde2 set


13/96

13

#et .nclusion

A" $xA implies thatx"

A " $ A" and " A

A" $ A" and A"

A is a subset of "

A and " are e4ual set

A is a proper subset of "


14/96

15

o6er set

All the possible subsets of a 7ien set X is call the

po6er set of X, denoted byP(X) A+ AX*

|P(X) + !n6hen +X+ n

Xa, b, c*

(X) , a, b, c, a, b*, b, c*, a, c*, X*


15/96

1per!tions on Cl!ssi2!l Sets

1nion7B @ 6 6 or 6 B

Intersection7B @ 6 6 and 6 B

Complement7

@ 6 6 , 6 4

4 " 1ni/ersal Set

Set Difference7

B @ 6 6 and 6 B

Set difference is also denoted )! 0 B


16/96

18

#et perations

Complement

/nion

.ntersection

difference


17/96

UNI1N 1F T1 SETS


18/96

G

INTE4SECTI1N 1F T1 SETS


19/96

;


20/96

2-

C156LE5ENT 1F A SET


21/96

2


22/96

22


23/96

6roperties o# Cl!ssi2!l Sets

B @ B B @ B

$B C% @ $ B% C$B C% @ $ B% C

$B C% @ $ B% $C%$B C% @ $ B% $C%

@

@

4 @ 44 @ @

@


24/96

6roperties o# Cl!ssi2!l Sets

If B C, ten C;e :organ:s 1aw

(#


25/96

!

&eal numbers

%otal orderin7$ a b

&eal a2is$ the set of real number (x;a2is)

.nteral$


26/96

2+

De 5or,!n7s l!8s


27/96

2H

Distri)uti/e la5s


28/96

2G

Asso2i!tive l!8s


29/96

Can be extended to n sets

?eneralied ;e :organ 1aw# #:

@@

"sing ( ) to 'eep original processing order


30/96

?eneralied ;!alit 1aw

@

@

"sing ( ) to 'eep original processing order


31/96

1aw of the excl!ded middle

# #:2 @

1aw of the Contradiction

# #:2

T'ese la*s are not true (orFu%%+ Sets,


32/96


33/96

''


34/96

'*

5!ppin, o# Cl!ssi2!l Sets to Fun2tions( 3elates set0teoretic forms to function teoretic representation of

information( &ore generall!, it can )e used to map elements or su)sets on one

uni/erse of discourse to elements or sets in anoter uni/erse

( Suppose 4 and 1 are t5o uni/erses of discourse $information%( If an element 6 is contained in 4 and corresponds to an element !

contained in , it is generall!termed as mapping from 4 to ,of fA 4 0?

. As a mapping, the characteristic (indicator) function

expresses membership in set A for the element x in the universe

This membership idea is a mapping from an element x in the universe X to

one of the two elements in universe Y i!e!, to the elements " or #$or an% set A defined on the universe X, there exists a function&theoretic set,

called a 'alue et, denoted b% '(A), under the mapping of the characteristic

function, x! % convention the null set is assigned the membershipvalue " and the whole set X is assigned the membership value #

= Ax

Axx

A if0

if1

)(


35/96

'J


36/96

'+


37/96

'H

Fuzzy T9eory

Fuzzy Fuzzy Un2ert!inty : 5!t9e(!ti2s De2ision;5!


38/96

'G

( fu! sets " the truth of a statement

becomes a matter of degree


39/96

';

( fu! mem)ersip functions


40/96

*-

( fu! mem)ersip functions$anoter e6ample%


41/96

*

De#initions 0 #uzzy sets

Fuzzy sets"

admits gradation suc as all tones )et5een)lac8 and 5ite. fu! set as a grapicaldescription tat e6presses o5 te transition fromone to anoter ta8es place. Tis grapicaldescription is called a mem)ersip function.

De#inition 7 Ket 4 )e some set of o)jects, 5itelements noted as 6.

" Tus, 4 @ 6. For e6ample, if 4 5ere to e:ualte set of all common ouse pets, ten

" 4 @ dogs, cats, fis, )irds, 5ere 6 @ dogs,62 @ cats, 6' @ fis, and 6* @ )irds.


42/96

Fuzzy Sets

Caracteristic function 4, indicating te )elongingness

of 6 to te set

4$6% @ 6 - 6

or called (e(bers9ip

9ence,B 4B$6%

@ 4$6% 4B$6%@ ma6$4$6%,4B$6%%

Note Some )oo8s use L for , )ut still it is not ordinar!additionM

Some more e6planations follo5


43/96

Fuzzy Sets

B 4B$6%

@ 4$6% 4B$6%

@ min$4$6%,4B$6%%

4$6% @ " 4$6%

B 4$6%


44/96

Fuzzy Sets

Note $6% O-,

not -, li8e Crisp set

@

$6% Q 6 L

$62% Q 62 L

@ $6i% Q 6i

Note7 RL add

RQ di/ide

#nl! for representing element and its

mem)ersip.

lso some )oo8s use $6% for Crisp Sets too.


45/96

Fuzzy Set 1per!tions

B$6% @ $6% B$6%

@ ma6$$6%, B$6%%

B$6% @ $6% B$6%

@ min$$6%, B$6%%

$6% @ 0 $6%

De &organs Ka5 also olds7$B% @ B$B% @ B

But, in general

X


46/96

*+

Fu! SetFu! Setis a set 5it a smoot )oundaries

Fu! Set Teor!Fu! Set Teor!generalies classical set teor!to allo5partial membership

Fu! Setis a uni/ersal set 4 is determined )! a

(e(bers9ip #un2tion$6%$6%tat assigns to eac

element 64 a num)er $6% in te unit inter/alO-,

1ni/ersal set 4 $Universe o# Dis2ourse% contains all

possi)le elements of concern for a particular

application Fu! set as a one0to0one correspondenceone0to0one correspondence

5it its mem)ersip function


47/96

*H

Fu! setis defined as

A = + &" A&'' )" /" A&'>$"%?

$6% @ Degree$6Degree$6%%is a ,r!de o# (e(bers9ipof

element 64 in set

/% /- /3 . $ %@- %

.. unit interv!l

N

.

.

U &universe o# dis2ourse'


48/96

*G

&em)ersip functions can )e represented &!'graphically,

&b'in a tabularor listform, &2 'analyticallyand &d'

geometrically$as a points in te unit 2ube%

eometrical representation for t5o0element uni/ersal set

U= &+%"-)'as a follo5ing /iualiation7 (e(bers9ip v!lues

%.$ &$"%' &%"%'

-

%

$.$ U &$"$'

% - % &%"$'

,r!p9i2!l &st!nd!rd' set o# (!i(u( #uzzinessB represent!tion #or(


49/96

*;

[see the previous figure] ertices $-,-%, $-,%,

$,-% and $,% represent !ll 2risp setstat can )e

defined for te uni/ersal set U, e.g. te point $,-%

corresponds to te crisp set +%) $element -as

no membership%

&em)ersip functions can )e symmetricalor

asymmetrical, and te most commonl! used forms

are triangulartriangular, trapeoidaltrapeoidal, aussianaussianand )ell)ell$te

first t5o dominate in applications due to si(pli2ity

and computational e##i2ien2y%

&em)ersip functions are typicallydefined on

one0dimensional uni/erses, and in most cases, te

mem)ersip function appears in te continuoscontinuos

formform


50/96

6roperties o# Fuzzy Sets

B @ B B @ B

$B C% @ $ B% C$B C% @ $ B% C

$B C% @ $ B% $C%$B C% @ $ B% $C%

@ @

4 @ 4 4 @ @ @

If B C, ten C

@


51/96

J

Te eigt of a fu! set eigt of a fu! set is te igest $ma6imum% /alue

of its mem)ersip function, i.e. 9ei,9t&A' =

If a fu! set as a 9ei,9t %, ten it is called a normal fu!normal fu!

setsetA in contrast, if eigt$% , te fu! set is said to )e

su)normalsu)normal

su)normal set is a fu! set tat contains onl! elements

5itpartial (


52/96

J2

set of all elements of te uni/ersal set U5it a propert!

A&' = %$ is a fu! set% is called te core of a fu! set core of a fu! set

$2ore&A'%

%

$

! b U = >!"b?

2ore&A'

supp&A'

9ei,9t&A' = % (normal fuzzy set)

5e(bers9ip#un2tion 9!s !

tr!pezoid!l #or(


53/96

J'

Fu! SetFu! Setis a set 5it a smoot )oundaries

Fu! Set Teor!Fu! Set Teor!generalies classical set teor!

to allo5partial membership

Fu! Setis a uni/ersal set 11is determined )! a

(e(bers9ip #un2tion$6%$6%tat assigns to eacelement 61 a num)er $6% in te unit inter/alO-,

1ni/ersal set 11$Universe o# Dis2ourse% contains all

possi)le elements of concern for a particular

application

Fu! set as a one0to0one correspondenceone0to0one correspondence

5it its mem)ersip function


54/96

J*

In#or(!tion orld

Crisp set as a uni:ue mem)ersip function

$6% @ 6

- 6

$6% -,

Fu! Set can a/e an infinite num)er of mem)ersip

functionsO-,


55/96

55

Fuzziness

E6amples7

num)er is close to J

Fuzziness


56/96

J+

Fuzziness

E6amples7

9eQse is tall


57/96

JH

E6ample7 =oung>

( E6ample7

" nn is 2G, -.G in set =oung>

" Bo) is 'J, -. in set =oung>

" Carlie is 2', .- in set =oung>

( 1nli8e statistics and pro)a)ilities, te degreeis not descri)ing

probabilitiestat te item is in te set, )ut instead descri)es to what

extent te item is te set.


58/96

JG

Fuzzy Subset

1 @ , 2, ', *,.,-

@ , 2, ', *, J

B @ 2, ', *

B in C3ISP SET T9E#3

$6% ?@ B$6%, 6

In terms of mem)ersip predicate

Crisp su)setood

S,$6% @ S2$6%, 6


59/96

J;

eo(etri2 Interpret!tion&%'

$-,%$,%

$-,-% $,-%

B

B2 B'

62

6

1 @ 6 , 62

Bis are suc tat

Bi$6% @ $6%, 6


60/96

+-

eo(etri2 Interpret!tion &-'

( Te points 5itin te !percu)e for 5ic is te upper

rigt corner are te su)sets of .

( Space defined )! te s:uare is te po5er set of .

( Formulation of ZADE, classical fu! set teor!

( For B to )e a su)set of , B$6% @ $6%, 6.

Tis means B P$% crispl!.

B


61/96

+

eo(etri2 Interpret!tion &3'

( Eac Bi is a su)set of to some degree.

BB2

B'

( 3esult of 1nion, Intersection, Complement is a SET


62/96

+2

5e(bers9ip #un2tion o# #uzzy lo,i2

ge2J *- JJ

Youn, 1ld

5iddle

-.J

D#&

Degree of

Membership

Fu! /alues

Fu! /alues a/e associated degrees of mem)ersip in te set.

-


63/96

+'

5e(bers9ip Fun2tions o# t9e e!(ple on !,e

% 89en = -

A%&' = &G$;'@% 89en - G$

$ 89en H=G$

$ 89en eit9er = - or H=

A-&' = &;-'@% 89en - G$ &;'@% 89en G$

$ 89en = G$

A3&' = &;G$'@% 89en G$

% 89en H=


64/96

+*

5e(bers9ip Fun2tions usin, Fuzzy Sets 0Z!de97s Not!tion

@ .-Q- L .-Q2J L -.JQ'2.J L -.-Q*-

& @ -.-Q2J L -.JQ'2.J L .-Q*- L -.JQ*H.J L -.-QJJ

# @ -.-Q*- L -.JQ *H.J L .-QJJ

Linguistic Values


65/96

Linguistic Values

>

embership

0

1

Heightshort medium tall

180 1?0 1@0 (cm)

>embership

0

1

Weightli7ht medium heay

0 ?0 90 (7)

Definitions7 Fu! Sets $figure from UlirVuan%


66/96

++

Definitions7 Fu! Sets $figure from UlirVuan%

&em)ersip functions $figure from UlirVuan%


67/96

+H

p $ g %

F set $fig re from Earl Co %


68/96

+G

Fu! set $figure from Earl Co6%

Fu! Set $figure from Earl Co6%


69/96

+;

Fu! Set $figure from Earl Co6%


70/96

H-

Fuzzy Set 1per!tions

1nion 7 )()(BBBB

xx BABA =

Intersection 7 )()(BBBB

xxBABA

=

Complement 7 )(1)(BB

xxAA

=


71/96

?1

Operations of Fuzzy SetsOperations of Fuzzy Sets

.ntersection /nion

Complement

ot A

A

Containment

A

B

BA A B


72/96

?!

ComplementComplement

Crisp #ets$ ho does not belon7 to the setD -uzzy #ets$ Eo6 much do elements not belon7 to the setD

%he complement of a set is an opposite of this setF -or

e2ample, if 6e hae the set of tall men, its complement is

the set of G% tall menF hen 6e remoe the tall men set

from the unierse of discourse, 6e obtain the complementF

.f A is the fuzzy set, its complement Acan be found asfollo6s$

A(x) 1 A(x)


73/96

?3

ContainmentContainment

Crisp #ets$ hich sets belon7 to 6hich other setsD -uzzy #ets$ hich sets belon7 to other setsD

#imilar to a Chinese bo2, a set can contain other setsF %he

smaller set is called the subsetF -or e2ample, the set of tall

men contains all tall menH ery tall men is a subset of tall

menF Eo6eer, the tall men set is Iust a subset of the set of

menF .n crisp sets, all elements of a subset entirely belon7

to a lar7er setF .n fuzzy sets, ho6eer, each element canbelon7 less to the subset than to the lar7er setF lements of

the fuzzy subset hae smaller memberships in it than in the

lar7er setF


74/96

?5

IntersectionIntersection

Crisp #ets$ hich element belon7s to both setsD -uzzy #ets$ Eo6 much of the element is in both setsD

.n classical set theory, an intersection bet6een t6o sets contains the

elements shared by these setsF -or e2ample, the intersection of the set

of tall men and the set of fat men is the area 6here these sets oerlapF

.n fuzzy sets, an element may partly belon7 to both sets 6ith different

membershipsF

A fuzzy intersection is the lower membershipin both sets of each

elementF %he fuzzy intersection of t6o fuzzy setsAandBon unierse

of discourse X$

AB(x) min


75/96

?

UnionUnion

Crisp #ets$ hich element belon7s to either setD -uzzy #ets$ Eo6 much of the element is in either setD

%he union of t6o crisp sets consists of eery element that falls into

either setF -or e2ample, the union of tall men and fat men contains all

men 6ho are tall & fatF

.n fuzzy sets, the union is the reerse of the intersectionF %hat is, the

union is the largest membership alue of the element in either setF

%he fuzzy operation for formin7 the union of t6o fuzzy sets A and "

on unierse X can be 7ien as$

A"(2) ma2


76/96

?8

Operations of Fuzzy SetsOperations of Fuzzy Sets

Complement

0x

1

(x)

0x

1

Containment

0x

1

0x

1

A B

Not A

A

.ntersection

0x

1

0x

A B

/nion

0

1

A B

A B

0x

1

0x

1

BA

B

A

(x)

(x) (x)


77/96

HH


78/96

HG

De 5or,!n7s L!8s

BBBB

BABA =

BBBB

BABA =


79/96

H;

6roperties o# Fuzzy Set

Commutati/it! BBBB ABBA =

BBBBABBA =

ssociati/it!BBBBBB

CBACBA =

BBBBBB CBACBA =

CABACBA


80/96

G-

Distri)uti/it! BBBBBBBCABACBA =

BBBBBBBCABACBA =

Idempotenc!BBB

AAA =BBBAAA =

Identit! BBAA =

=B

A

XXA =B

BBAXA =


81/96

G

Transiti/it!BBBBB

CAthenCBAIf

In/olutionBB

AA =

Artificial .ntelli7ence ' C#385Artificial .ntelli7ence ' C#385-uzzy Lo7ic-uzzy Lo7ic


82/96

@!

!uality!uality

-uzzy setAis considered e4ual to a fuzzy setB, .- AGJGLK .- (iff)$

A(x) B(x), xX

A 0F31 M 0F! M 13

B 0F31 M 0F! M 13

thereforeAB



83/96

@3

InclusionInclusion

.nclusion of one fuzzy set into another fuzzy setF -uzzy setA Xis included in (is a subset of) another fuzzy set,B X$

A(x) B(x), xX

ConsiderX 1, !, 3* and setsAandB

A 0F31 M 0F! M 13H

B 0F1 M 0F! M 13

thenAis a subset ofB, or AB



84/96

@5

Car"inalityCar"inality

Cardinality of a non;fuzzy set, N, is the number of elements in NF"/% the cardinality of a fuzzy set A, the so;called #.O>A C/G%, is

e2pressed as a #/> of the alues of the membership function of A,

A(x)$

cardA A(x1) M A(x!) M P A(xn) QA(xi), for i1FFn

ConsiderX 1, !, 3* and setsAandB

A 0F31 M 0F! M 13H

B 0F1 M 0F! M 13

cardA 1F@

cardB !F0



85/96

@

mpty Fuzzy Setmpty Fuzzy Set

A fuzzy setAis empty, .- AGJ GLK .-$A(x) 0, xX

ConsiderX 1, !, 3* and setA

A 01 M 0! M 03

thenAis empty

E6ample7 n uni/erse as tree elements, 4@ a,),c . We desire to


86/96

G+

map te elements of te po5er set of 4, i.e., P$4%, to a uni/erse, ,

consisting of onl! t5o elements $ te caracteristic function% @ -,

Te elements of te po5er set are enumerated as follo5s7

P$4% @

" a, ), c, a,), ),c, $a,c, $a,),c

Tus, te elements in te /alue set $% as determined from te

mappings are

P$4% @ -,-,-, ,-,-, -,,-, -,-,, ,,-, -,,, ,-,,,,

For e6ample, te tird su)set in te po5er set P$4% is te element ).

For tis su)set tere is no a, so a /alue of - goes in te first

position of te data tripletA tere is a ), so a /alue of goes in te

second position of te data tripletA and tere is no c, so a /alue of -goes in te tird position of te data triplet. 9ence, te tird su)set

of te /alue set is te data triplet -,,-, as alread! seen.


87/96

GH

E!(ple

+++=

+++=

5F

5!F

3?F

!F

!F

53F

3F

!1

BBBA

++++=

++++=

8F

5

@F

3

3F

!

F

1

1

@F

5

?F

3

F

!

0

1

1

BB

BA

+++=

5F

5

3F

3

?F

!

1

BBBA

+++=

!F

5

!F

3

F

!

F

BBBA

!33


88/96

GG

+++==

!F

5

3F

3

3F

!

F+

BBBBBABA

+++==

5F

5

!F

3

F

!

0+BBBB

ABAB

++++==

8F

5

?F

3

3F

!

0

1

1

BBBB BABA

++++==

@F

5

@F

3

F

!

F

1

1

BBBBBABA


89/96

G;

++++=

@F

5

?F

3

F

!

1

1

1

BBAA

+++=

5F

5

!F

3

3F

!

F

BBBB

E l f t ti


90/96

;-

E6ample fu! set operationsA:

A" A"

A "

A

#perations on Fu! Sets


91/96

;

( Sligtl! differs from regular set operations

Fu! inter/al )et5een J V G Fu! num)er a)out *

AND" 14" NEATI1N


92/96

;2

" "

Canalso )e called intersect, unif!, and negate

ND #3 NETI#N

oung emplo!ees salar!


93/96

;'

g p ! !( Xuestion7 5at is a !oung emplo!ees salar!

Documents

CLASSICAL AND FUZZY SETS-VTU.ppt