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FuzzyLogic
PremKalraDepartmentofComputerScience
andEngineringIITDelhi
Boolean(Crisp)Logic
• Isitraining– Yes/No– True/FalseCrispanswerIfanswer–maybe,maynotbe,…(vague)Notacrispanswer(fuzzyanswer)
FuzzyLogic
• Isthispersontall?– Yes/No– Verytall,quitetall,notsotall,nottall,
7’ 6’ 5’6” 5’BeSer(moreprecise)descripTon!IntroducedbyZadeh(1965),UniversityofCaliforniaBerkley
FuzzyLogic
� ApproximaTon(“granulaTon”)AcolorcanbedescribedpreciselyusingRGBvalues,oritcanbeapproximatelydescribedas“red”,“blue”,etc.� Degree(“graduaTon”)Twodifferentcolorsmaybothbedescribedas“red”,butoneisconsideredtobemoreredthantheother
� FuzzylogicaSemptstoreflectthehumanwayofthinking
FuzzySet
X=EnTrepopulaToninaclassM=AllmalepopulaTon(m1,m2,m3,…,mN)F=AllfemalepopulaTon(f1,f2,f3,…,fL)CrispSets
M F
FuzzySet
X=EnTrepopulaToninaclassofSIV895/CSM802S=AllgoodstudentsS={(s,g)|sεX}andg(s)isameasurementofgoodnessofstudents.Forexample:S={(Rakesh,0.8),(Sunita,0.7),(Farhan,0.1),(Joseph,0.9)}etc
FuzzySetCrispSet FuzzySetS={s|sεX} F={(s,µ)|sεX}
µ(s)isthedegreeofs.ItisacollecTonofelements ItisacollecTonofordered
pairsInclusioniscrisp(yesorno) Inclusionisfuzzy,i.e.ifpresent
thenwithadegreeofmembership
Crispsetisafuzzysetwithextrememembershipvalues(0or1).
RelatedTermsFuzzyrelaTonRelaTonshipscanalsobeexpressedonascaleof0to1e.g.degreeofresemblancebetweentwopeopleFuzzyvariableVariablewith(labelsof)fuzzysetsasitsvaluesLinguisTcvariableFuzzyvariablewithvaluesthatarewordsorsentencesinalanguagee.g.variablecolorwithvaluesred,blue,yellow,green…LinguisTchedgeTermusedasamodifierforbasictermsinlinguisTcvaluese.g.wordssuchasvery,abit,rather,somewhat,etc.
FuzzySet
ExamplesFuzzySetIfcoldisafuzzyset,exacttemperaturevaluesmightbemappedtothefuzzysetasfollows:• 15degrees→0.2(slightlycold)• 10degrees→0.5(quitecold)• 0degrees→1(extremelycold)
MembershipFuncTonIfXisauniverseofdiscourseandxεX,thenafuzzysetAinXisdefinedasasetoforderedpairs,thatisA={(x,µΑ(x))|xεX}where,µΑ(x)iscalledthemembershipfuncTonforthefuzzysetA.Note:µΑ(x)mapeachelementofXontoamembershipgrade(ormembershipvalue)between0and1(bothinclusive).
MembershipFuncTonAfuzzysetiscompletelycharacterizedbyitsmembershipfuncTon.So,itwouldbeimportanttolearnhowamembershipfuncToncanbeexpressed(mathemaTcallyorotherwise).Note:AmembershipfuncToncanbeon(a)adiscreteuniverseofdiscourseand(b)aconTnuousuniverseofdiscourse.
MembershipFuncTonNote:AmembershipfuncToncanbeon(a)adiscreteuniverseofdiscourseand(b)aconTnuousuniverseofdiscourse.
MembershipFuncTonSingleton MF
MembershipFuncTonRectangular MF
MembershipFuncTonTriangular MF
MembershipFuncTonTrapeziodal MF
MembershipFuncTonGaussian MF
MembershipFuncTonBell (Cauchy) MF
MembershipFuncTonSigmoidal MF
ExampleCourse Grading (Crisp)
E
ExampleCourse Grading (Fuzzy)
E
OperaTonsonFuzzySets
OperaTonsonFuzzySets
OperaTonsonFuzzySets
ProperTes
ProperTes
A
OperaTonsonFuzzySetsExample
OperaTonsonFuzzySetsExample
OperaTonsonFuzzySetsExample
FuzzyvsProbability
Fuzzy:WhenwesayaboutcertaintyofathingExample:ApaTentcometothedoctorandhehastodiagnosesothatmedicinecanbeprescribed.Doctorprescribedamedicinewithcertainty60%thatthepaTentissufferingfromflue.So,thediseasewillbecuredwithcertaintyof60%anduncertainty40%.Here,insteadofflue,otherdiseaseswithsomeothercertainTesmaybe.Probability:WhenwesayaboutthechanceofaneventtooccurExample:IndiawillwintheT20tournamentwithachance60%meansthatoutof100matches,Indiaown60matches.
FuzzySystem
OperaTonsonFuzzySetsExercise
