(英文PPT4)模糊控制讲义第二章(2.5 2.6)

CHAPTER2FuzzyMathematics

2.5FuzzyMatrix2.6FuzzyRelation2.5FuzzyMatrix2.5.1Basicdefinitionsoffuzzymatrix2.5.2Supportoffuzzymatrix2.5.3Compositionoffuzzymatrix2.5.4Transposeoffuzzymatrix2.5.1BasicdefinitionsoffuzzymatrixIf(,

),thereexist,thenisafuzzymatrix.Usually,

meansallthefuzzymatrixes(rows,

lines).1Union:

2Intersection:

2.5.1Basicdefinitionsoffuzzymatrix3Complement:

4Contain:5Equal:2.5.1BasicdefinitionsoffuzzymatrixExample2.5.1:

Notes:Afuzzymatrixanditscomplementarenotcomplementaryeventssimilarwithfuzzysets.Example2.5.2:

2.5.1BasicdefinitionsoffuzzymatrixExample2.5.3:

2.5.2SupportoffuzzymatrixDefinition:

iscalledthesupportmatrixof.Notes:ThesupportmatrixisaBooleanmatrix.2.5.2SupportoffuzzymatrixExample2.5.4:

2.5.3CompositionoffuzzymatrixDefinition:,,,,thecompositionmatrixofthesetwomatrixesisafuzzymatrixwithrowsandlines,writtenaswhoseelementisThecompositionoffuzzymatrixisalsocalledasmultiplicationoffuzzymatrix.2.5.3CompositionoffuzzymatrixExample2.5.5:

Notes:(popularly)

2.5.4TransposeoffuzzymatrixRowschangeintolinesandlineschangeintorows.2.6FuzzyRelation2.6.1Definitionoffuzzyrelation2.6.2Operationoffuzzyrelation2.6.3Compositionoffuzzyrelation2.6.1DefinitionoffuzzyrelationRelation:AsubsetofaCartesianproductofand,iscalledadualisticrelationfromto(orrelationforshort).Fuzzyrelation:andaretwosets,isafuzzysubsetin,isalsocalledafuzzyrelationfrom

to

.Thedegreeofmembershipof

is2.6.1DefinitionoffuzzyrelationFuzzyrelationcandescribedbyfuzzymatrixwhenandarefiniteset.When,itiscalledthefuzzyrelationin.Anelementinfuzzymatrixmeanstherelationbetweenthe

element

inandtheelementin.2.6.1Definitionoffuzzyrelationisthespaceofheightinanareaand

isthespaceofweight.

,(cm),

,(kg),therelationbetweenheightandweightisintable.

Example2.5.62.6.1Definitionoffuzzyrelation405060708014010.80.20.101500.810.80.20.11600.20.810.80.21700.10.20.810.818000.10.20.81

Fuzzyrelationbetweenheightandweight2.6.1DefinitionoffuzzyrelationWecanfindfromtheaboveexamplethathowdoesfuzzymathematicsdescribethefuzzyconceptioninrealworldintheformofmath.Usingthismethod,wecanhandlethefuzzyconceptionincomputer.2.6.2Operationoffuzzyrelation

andaretwofuzzyrelationsin.1Union:2Intersection:3Contain:4Equal:5Complement:6Transpose:2.6.2OperationoffuzzyrelationCharacteristicsoffuzzyrelation:

1自反性2对称性

Everyelementindiagonaloffuzzymatrixis1.

3传递性2.6.2OperationoffuzzyrelationExample2.6.1Twopeoplehavethefuzzyrelations:“similitude”“enemy”“love”“younger”自反性（Y）对称性（Y）传递性（N）自反性（N）对称性（Y）传递性（N）自反性（Y）对称性（?）传递性（?）自反性（N）对称性（N）传递性（Y）2.6.2OperationoffuzzyrelationExample2.6.2自反性（Y）对称性（Y）传递性（N）2.6.3Compositionoffuzzyrelation

isaspaceofpeople,isthebrotherrelation,isthepaternity,istherelationbetweenuncleandnephew.,,arethreepeoplein,isabrotherof,isthefatherof,soshouldbetheuncleof.Wecalltherelationbetweenuncleandnephewisthecompositionbetweenthebrotherrelationandpaternity.Andwrittenas：Ifthen2.6.3CompositionoffuzzyrelationInageneralway,

，，isthespace,isacommonrelationfrom

to,isacommonrelationfrom

to

,isacommonrelationform

to,ifiscalledacompositionfrom

to,writtenas.2.6.3CompositionoffuzzyrelationDefinition:Let

，，bethespace,

isafuzzyrelationfrom

to

,

isafuzzyrelationfrom

to

,isafuzzyrelationformto,isthecompositionfromtowhohasthemembershipfunction:When

，，isfinite,thecompositi