1. Introduction • Preliminaries • Some Useful Definitions • Types of fuzzy sets • Degree of Fuzzy Sets • Operators of Fuzzy Sets • Conditions & Limitations • Multiplication • Summation • Operators of Theory Sets • Characteristics of S & T • Some definitions for T & S • Unity and Community Defs. • Mean Operators • Fuzzy AND & OR • Combinations of Fuzzy AND & OR 2. Fuzzy Measurement & Measurement of Fuzzy Sets • Fuzzy Measurement • Dr. ASGARI Zadeh Possibility Definition • SUGENO Definition • Possibility Definition • Graph of S Function • Measurement of Fuzzy Sets • Entropy of Fuzzy Sets (De Luca & Termini) • YAGER Definition for à 3. Propagation principle • Propagation principle & Applications • Propagation principle and Second Types of Fuzzy Sets • Fuzzy Numbers & Algebraic Operations • Fuzzy Numbers Intervals • L-R Interval Function (Asymmetric) • L-R Interval Function • L-R Interval Function Operations 4. Functions & Fuzzy Analyzing • Functions & Fuzzy Analyzing • Functions & Fuzzy Analyzing • Fuzzy functions Extremes • Integral of Fuzzy Functions • Integral of Type 2 fuzzy function with definite interval • Differentiation of Definite functions With Fuzzy Domains & Ranges • Integral of fuzzy function with definite interval • Properties of fuzzy Integral • Integral of Definite functions with fuzzy interval 5. Relations & Fuzzy Graphs • Fuzzy Relations • Fuzzy Graphs in Fuzzy Sets. • Fuzzy Images in 2-D Graphs • Fuzzy Images in n-D Graphs • Operations in Fuzzy Graphs • Fuzzy Forests

1. In The Name of GodIn The Name of God
Advance MathematicsAdvance Mathematics

2. Theory of Fuzzy SetsTheory of Fuzzy Sets
Amir RafatiAmir Rafati
8612100486121004

3. Computerizing DecisionComputerizing Decision

4. Chapter 1:Chapter 1:
IntroductionIntroduction

5. IntroductionIntroduction
 PragmatismPragmatism
 AmbiguityAmbiguity
 FuzzyFuzzy

6. PreliminariesPreliminaries
Forms of fuzzy sets:Forms of fuzzy sets:
Type I:Type I:
Normal Sets:Normal Sets: A={2,3,4}A={2,3,4}
Fuzzy Sets:Fuzzy Sets: Ã={(1,0),(2,1),(3,1),(4,1),(7,0)}Ã={(1,0),(2,1),(3,1),(4,1),(7,0)}
Or Ĩ ={(2,0.7),(9,1),(8,0),(4,0.3)}Or Ĩ ={(2,0.7),(9,1),(8,0),(4,0.3)}
Type II:Type II:
Normal Sets:Normal Sets: A={X| 1<X<3}A={X| 1<X<3}
Fuzzy Sets:Fuzzy Sets: Ã={(x,Ã={(x, μμÃà (x))| x(x))| xЄЄX}X}
((μμÃà : Member function): Member function)

7. Some Useful Definitions:Some Useful Definitions:
 Fuzzy Support Set: S(Ã)Fuzzy Support Set: S(Ã)
Ĩ ={(2,0.7),(9,1),(8,0),(4,0.3),(7,0),(1,0.2)} => S(Ĩ)={1,2,9,4}Ĩ ={(2,0.7),(9,1),(8,0),(4,0.3),(7,0),(1,0.2)} => S(Ĩ)={1,2,9,4}
Fuzzy Set in stage ofFuzzy Set in stage of αα: A: Aαα
AAαα ={x={xЄЄX|X| μμà (x)≥à (x)≥ αα} =>} => AA0.70.7 = {2,9}= {2,9} (Strong(Strong A0.7:A0.7: A’0.7={9}A’0.7={9}))
 Main number of finite à is shown by: |Ã|Main number of finite à is shown by: |Ã|
|Ã|=∑|Ã|=∑ μμà (x)à (x)
 Relative Main number is shown by: ||Ã||Relative Main number is shown by: ||Ã||
||Ã||= |Ã|/|X|||Ã||= |Ã|/|X|
Example:Example:
|Ã|=0.7+1+0.3+0.2=2.2|Ã|=0.7+1+0.3+0.2=2.2
||Ã||=2.2/5=0.44||Ã||=2.2/5=0.44

8. Some Useful Definitions:Some Useful Definitions:
 Two fuzzy Sets are Equal if & only if:Two fuzzy Sets are Equal if & only if:
 if and only if:if and only if:
Example:Example:
Ã={(1,0.2),(2,0.8),(4,0)} & Ĩ ={(1,0.2),(2,1),(3,0.3)}Ã={(1,0.2),(2,0.8),(4,0)} & Ĩ ={(1,0.2),(2,1),(3,0.3)}
 SUP Ã means upper limit of :SUP Ã means upper limit of :
Hgt (Ã)= SupHgt (Ã)= Sup
Example: Ã={(1,0.2),(2,0.8),(4,0)} => hgt (Ã)= 0.8Example: Ã={(1,0.2),(2,0.8),(4,0)} => hgt (Ã)= 0.8

9. Types of fuzzy sets:Types of fuzzy sets:
 S function:S function:  Z functionZ function

10. Types of fuzzy sets:Types of fuzzy sets:
 ππ functionfunction  V functionV function

11. Degree of Fuzzy SetsDegree of Fuzzy Sets
 Fuzzy Sets with Degree of ONE:Fuzzy Sets with Degree of ONE:
Ĩ ={(2,0.7),(9,1),(8,0),(4,0.3)}Ĩ ={(2,0.7),(9,1),(8,0),(4,0.3)}
 Fuzzy Sets with Degree of TWO:Fuzzy Sets with Degree of TWO:
Ĩ ={(2,Ĩ ={(2,μμĨ (2)),(9,Ĩ (2)),(9, μμĨ (9)),(4,Ĩ (9)),(4, μμĨ (4))}Ĩ (4))}
μμĨ (2)={uĨ (2)={uii ,, μμ(u(uii )| i=1,..,3})| i=1,..,3}
μμĨ (2)={(0.6,0.9),(0.7,1),(0.9,0.4)}Ĩ (2)={(0.6,0.9),(0.7,1),(0.9,0.4)}
 Fuzzy Sets with Higher Degrees:Fuzzy Sets with Higher Degrees:
Ĩ ={(2,Ĩ ={(2,μμĨ (2)),(9,Ĩ (2)),(9, μμĨ (9)),(4,Ĩ (9)),(4, μμĨ (4))}Ĩ (4))}
μμĨ (2)={uĨ (2)={uii ,, μμ(u(uii )| i=1,..,3})| i=1,..,3}
μμĨ (2)={(0.6,Ĩ (2)={(0.6, μμĨ (0.6)),(0.7,Ĩ (0.6)),(0.7, μμĨ (0.7)),(0.9,Ĩ (0.7)),(0.9, μμĨ (0.9))}Ĩ (0.9))}
μμĨ (0.6)={(0.4,0.1),(0.9,1),(1,0.9)}Ĩ (0.6)={(0.4,0.1),(0.9,1),(1,0.9)}

12. Operators of Fuzzy Sets:Operators of Fuzzy Sets:
There is no-unique definition forThere is no-unique definition for
operators of fuzzy sets andoperators of fuzzy sets and
their definitions are related totheir definitions are related to
their application but one simpletheir application but one simple
definition will be introduced indefinition will be introduced in
this sectionthis section

13. Operators of Fuzzy Sets:Operators of Fuzzy Sets:
Ễ={(1,0.1),(2,0.3),(3,0.4)}Ễ={(1,0.1),(2,0.3),(3,0.4)}
Ễ={(1,0.5),(2,0.8),(3,0.4),Ễ={(1,0.5),(2,0.8),(3,0.4),
(4,0.9)}(4,0.9)}
Ã={(1,0.5),(2,0.8),(3,0.4)}Ã={(1,0.5),(2,0.8),(3,0.4)}
ĨĨ ={(1,0.1),(2,0.3),(3,0.5),(4,0.9)}={(1,0.1),(2,0.3),(3,0.5),(4,0.9)}
~Ã={(1,0.5),(2,0.2),(3,0.6)}~Ã={(1,0.5),(2,0.2),(3,0.6)}

14. Graphical Example:Graphical Example:

15. Conditions & LimitationsConditions & Limitations
μμ(S and T)=f((S and T)=f(μμs,s, μμT)T)
μμ(S or T)=g((S or T)=g(μμs,s, μμT)T)
 μμS ^S ^ μμT=T= μμT ^T ^ μμSS
 μμS vS v μμT=T= μμT vT v μμSS
 ((μμS ^S ^ μμT) ^T) ^ μμQ =Q = μμS ^ (S ^ (μμT ^T ^ μμQ)Q)
 ((μμS vS v μμT) vT) v μμQ =Q = μμS v (S v (μμT vT v μμQ)Q)
 ((μμS ^S ^ μμT) vT) v μμQ = (Q = (μμS vS v μμQ) ^ (Q) ^ (μμT vT v μμQ)Q)
 ((μμS vS v μμT) ^T) ^ μμQ = (Q = (μμS ^S ^ μμQ) v (Q) v (μμT ^T ^ μμQ)Q)
 ((μμS ^S ^ μμT) ≤ Min(T) ≤ Min(μμS,S,μμT)T)
 ((μμS vS v μμT) ≥ Max(T) ≥ Max(μμS,S,μμT)T)
 1 ^ 1=11 ^ 1=1
 0 v 0=00 v 0=0

16. Multiplication:Multiplication:
×:×:
D(Ã1, Ã2,…, Ãn)={X1,X2,…,Xn}D(Ã1, Ã2,…, Ãn)={X1,X2,…,Xn}
μμ(Ã1×Ã2×…×Ãn)=Min(i) {(Ã1×Ã2×…×Ãn)=Min(i) {μμÃi(xi)| x=(x1,x2,…,xn), xiÃi(xi)| x=(x1,x2,…,xn), xiЄЄ Xi}Xi}
μμ (Power (Ã (x), m)= Power ((Power (Ã (x), m)= Power (μμ (Ã (x)), m)(Ã (x)), m)

17. Summation:Summation:
 Ễ=Ã+Ĩ:Ễ=Ã+Ĩ:
Ễ={(x,Ễ={(x, μμÃ+ĨÃ+Ĩ (x))| x(x))| xЄЄX}X}
μμÃ+ĨÃ+Ĩ (x)=(x)= μμÃà ++ μμĨĨ –– μμà *à * μμĨĨ
 Ễ=Ã + Ĩ (finite Sum)Ễ=Ã + Ĩ (finite Sum)
Ễ={(x,Ễ={(x, μμà + Ĩà + Ĩ (x))| x(x))| xЄЄX}X}
μμà + Ĩà + Ĩ (x)= Min((x)= Min(μμÃà ++ μμĨĨ,1),1)

18. Finite Difference & Dot Product:Finite Difference & Dot Product:
 Ễ=ÃỄ=Ã ΘΘ Ĩ:Ĩ:
Ễ={(x,Ễ={(x, μμÃà ΘΘ Ĩ (x))| xĨ (x))| xЄЄX}X}
μμÃà ΘΘ Ĩ (x)=Max (Ĩ (x)=Max (μμà +à + μμĨ – 1,0)Ĩ – 1,0)
 Ễ=Ã . Ĩ:Ễ=Ã . Ĩ:
Ễ={(x,Ễ={(x, μμà . Ĩ (x))| xà . Ĩ (x))| xЄЄX}X}
μμà . Ĩ (x)=à . Ĩ (x)=μμà .à . μμĨĨ

19. An Example:An Example:
à (x)={(3,0.5),(5,1),(7,0.6)}à (x)={(3,0.5),(5,1),(7,0.6)}
Ĩ (x)={(3,1),(5,0.6)}Ĩ (x)={(3,1),(5,0.6)}
 Ã × Ĩ = {[(3;3),0.5], [(5;3),1.0], [(7;3),0.6],Ã × Ĩ = {[(3;3),0.5], [(5;3),1.0], [(7;3),0.6],
[(3;5),0.5], [(5;5),0.6], [(7;5),0.6]}[(3;5),0.5], [(5;5),0.6], [(7;5),0.6]}
 Ã × Ã = {(3,0.25),(5,1),(7,0.36)}Ã × Ã = {(3,0.25),(5,1),(7,0.36)}
 Ã + Ĩ = {(3,1),(5,1),(7,0.6)}Ã + Ĩ = {(3,1),(5,1),(7,0.6)}
 Ã + Ĩ = {(3,1),(5,1),(7,0.6)}Ã + Ĩ = {(3,1),(5,1),(7,0.6)}
 ÃÃ ΘΘ Ĩ = {(3,0.5),(5,0.6)}Ĩ = {(3,0.5),(5,0.6)}
 Ã . Ĩ = {(3,0.5),(5,0.6)}Ã . Ĩ = {(3,0.5),(5,0.6)}

20. Operators of Theory Sets:Operators of Theory Sets:

21. Characteristics of S & T:Characteristics of S & T:
 t(0,0)=0 ; t(1,t(0,0)=0 ; t(1, μμÃ)=Ã)= μμà ,xà ,x ЄЄ XX
 MonotonocityMonotonocity
IfIf μμà ≤à ≤ μμĨĨ andand μμỄ ≤Ễ ≤ μμÑÑ thenthen
t(t(μμà ,à , μμỄỄ )) ≤≤ t(t(μμĨ ,Ĩ , μμÑÑ ))
 CommutativityCommutativity
t(t(μμà ,à , μμỄỄ )) == t(t(μμỄ ,Ễ , μμÃà ))
 AssociativityAssociativity
t(t(μμà ,à , t(t(μμĨ ,Ĩ , μμÑÑ ))= t(t())= t(t(μμĨ ,Ĩ , μμÃà ),), μμÑÑ ))

22. Characteristics of S & T:Characteristics of S & T:
 S(1,1)=1 ; S(0,S(1,1)=1 ; S(0, μμÃ)=Ã)= μμà ,xà ,x ЄЄ XX
 MonotonocityMonotonocity
IfIf μμà ≤à ≤ μμĨĨ andand μμỄ ≤Ễ ≤ μμÑÑ thenthen
S(S(μμà ,à , μμỄỄ )) ≤≤ S(S(μμĨ ,Ĩ , μμÑÑ ))
 CommutativityCommutativity
S(S(μμà ,à , μμỄỄ )) == S(S(μμỄ ,Ễ , μμÃà ))
 AssociativityAssociativity
S(S(μμà ,à , S(S(μμĨ ,Ĩ , μμÑÑ ))= S(S())= S(S(μμĨ ,Ĩ , μμÃà ),), μμÑÑ ))

23. Some definitions for T & S:Some definitions for T & S:


 =µµµµ
=µµ


 =µµµµ
=µµ
µµ=µµµµ=µµ
µ−=µµ−µ−−=µµ
Otherwise1
0)}x(),x({Minif)}x(),x({Max
))x(),x((S
:SummationStrong
Otherwise0
1)}x(),x({Maxif)}x(),x({Min
))x(),x((t
:tionMultiplicaStrong
)))x((n)),x((n(t(n))x(),x((S)))x((n)),x((n(S(n))x(),x((t
:kerDec&Bonissone
)x(1))x((n))x(1),x(1(S1))x(),x((t
:asinAl
~~~~
~~
~~~~
~~
~~~~~~~~
~~~~~~
BABA
BA
w
BABA
BA
w
BABABABA
AABABA

24. Some definitions for T & S:Some definitions for T & S:
)x().x())x(),x((SSummationebraiclgA
)x()x())x(),x((ttionMultiplicaebraiclgA
)x()x(1
)x()x(
))x(),x((S
)]x().x()x()x([2
)x().x(
))x(),x((t
:TandSEinstein
)}x()x(,1{Min))x(),x((S
:SummationFinite
}1)x()x(,0{Max))x(),x((t
DifferenceFinite
~~~~
~~~~
~~
~~
~~
~~~~
~~
~~
~~~~
~~~~
BABA
2
BABA
2
BA
BA
BA
5.1
BABA
BA
BA
5.1
BABA
1
BABA
1
µµ=µµ
µ+µ=µµ
µ+µ+
µ+µ
=µµ
µµ−µ+µ−
µµ
=µµ
µ+µ=µµ
−µ+µ=µµ

25. Some definitions for T & S:Some definitions for T & S:
))x(),x((Max))x(),x((S))x(),x((S
))x(),x((Min))x(),x((t))x(),x((t
)}x()x(,1{Max))x(),x((S:Maximum
}1)x()x(,0{Min))x(),x((t:Minimum
)x().x())x(),x((SSummationebraiclgA
)x()x())x(),x((ttionMultiplicaebraiclgA
)x().x(1
)x().x(2)x()x(
))x(),x((S
)x().x()x()x(
)x().x(
))x(),x((t
:TandSHamacher
~~~~~~
~~~~~~
~~~~
~~~~
~~~~
~~~~
~~
~~~~
~~
~~~~
~~
~~
BABA
3~1
BA
w
BABA
3~1
BA
w
BABA
1
BABA
1
BABA
2
BABA
2
BA
BABA
BA
5.2
BABA
BA
BA
5.2
µµ≥µµ≥µµ
µµ≤µµ≤µµ
µ+µ=µµ
−µ+µ=µµ
µµ=µµ
µ+µ=µµ
µµ−
µµ−µ+µ
=µµ
µµ−µ+µ
µµ
=µµ

26. Unity and Community Defs.:Unity and Community Defs.:
1p},)(,1{Min)x()}Xx|)x(,x{(BA
1p},))1()1((,1{Min1)x()}Xx|)x(,x{(BA
:Yager
1,
)x().x(1
)x()x()x().x().1(
)x()}Xx|)x(,x{(BA
0,
))x().x()x()x()(1(
)x().x(
)x()}Xx|)x(,x{(BA
:Hamacher
p/1p
B
p
ABABA
~~
p/1p
B
p
ABABA
~~
BA
BABA
BABA
~~
BABA
BA
BABA
~~
~~~~~~
~~~~~~
~~
~~~~
~~~~
~~~~
~~
~~~~
≥µ+µ=µ∈µ=∪
≥µ−+µ−−=µ∈µ=∩
−≥℘
µµ℘+
µ+µ+µµ−℘
=µ∈µ=∪
≥∂
µµ−µ+µ∂−+∂
µµ
=µ∈µ=∩
∪∪
∩∩
∪∪
∩∩

27. Unity and Community Defs.:Unity and Community Defs.:
]1,0[,
}),1(),1{(Max
)}1(,,{Min)x().x()x()x(
)x(
)}Xx|)x(,x{(BA
]1,[)x().x(},{Min
],0[)x().x(
)x().x(
)x(
:DefinitionClearMore
]1,0[,
}),x(),x({Max
)x().x(
)x()}Xx|)x(,x{(BA
:adePrandDubois
~~
~~~~~~
~~
~~
~~~~
~~
~~
~~
~~
~~
~~~~
BA
BABABA
BA
BA
~~
BABA
BA
BA
BA
BA
BA
BABA
~~
∈α
αµ−µ−
α−µµ−µµ−µ+µ
=µ
∈µ=∪





α∈µµµµ
α∈µµ
α
µµ
=µ
∈α
αµµ
µµ
=µ∈µ=∩
∪
∪
∩
∩∩

28. Mean Operators:Mean Operators:

29. Fuzzy AND & OR:Fuzzy AND & OR:
]1,0[,,
2
)1(},{.
:
]1,0[,,
2
)1(},{.
:
~~
~~
~~
~~
~~
~~
))(),((
))(),((
∈∂∈
+
∂−+∂=
∈∂∈
+
∂−+∂=
XxMax
ORWerner
XxMin
ANDWerner
BA
BA
xxor
BA
BA
xxand
BA
BA
µµ
µµµ
µµ
µµµ
µµ
µµ

30. Fuzzy AND & OR:Fuzzy AND & OR:
ityCommutativ
tyMonotonici
Demorgan
Xxxx
ANDZysnoZimmerman
m
i
i
m
i
ix
CompAi
−
−
−
∈∂∈





−−





=
∂
=
∂−
=
∏∏ ]1,0[,,))(1(1)(
:&
1
)1(
1
))((
,
~
µµµ µ

31. Combinations of Fuzzy AND & OR:Combinations of Fuzzy AND & OR:
]1,0[,],.)[1(.
&:2#
]1,0[,},,{).1(},{.
&:1#
~~~~~~
~~
~~~~
~~
))(),((
))(),((
∈∂∈−+∂−+∂=
∈∂∈∂−+∂=
Xx
nCombinatioAdditionaltionMultiplicaExample
XxMaxMin
nCombinatioMaxMinExample
BABABA
xx
BABA
xx
BA
BA
µµµµµµµ
µµµµµ
µµ
µµ

32. Chapter 2:Chapter 2:
Fuzzy Measurement &Fuzzy Measurement &
Measurement of FuzzyMeasurement of Fuzzy
SetsSets

33. Fuzzy MeasurementFuzzy Measurement
 Sugeno Definition:Sugeno Definition:
If B is considered as a Sub Set of aIf B is considered as a Sub Set of a
Reference Set, a function can beReference Set, a function can be
defined asdefined as Fuzzy MeasurementFuzzy Measurement in thein the
interval of B if:interval of B if:
)lim()(lim...,:3
)()(,,:2
1)(,0)(:1
321 AgAgThenAAABA
BgAgThenBABA
Xgg
nn
n
∞→∞→
=⊆⊆⊆∈
≤⊆∈
==
β
φ

34. Fuzzy MeasurementFuzzy Measurement
 Dr. Asgari Zadeh Possibility(Dr. Asgari Zadeh Possibility())
Definition:Definition:
XxxxfXAxfSupA
XffunctionaasDefinedbecanDefinitionAbove
IndexSetasDefinedisIASupA
BAThenBAXBA
X
Ax
i
IiIi
i
⊂∀=⊂=
→
=
≤⊆∈
==
∏∏
∏∏
∏∏
∏∏
∈
∈∈
,})({)(),()(
:]1,0[:
,)()(:3
)()(,,:2
1)(,0)(:1

φ

35. Ex. Of Possibility(Ex. Of Possibility() Definition:) Definition:
Possibility(Possibility() of nearing x to 8:=) of nearing x to 8:=
Possibility(Possibility() of A has a member near to 8:=) of A has a member near to 8:=
∏ })({x
x 0 1 2 3 4 5 6 7 8 9 10
0 0 0 0 0 0.1 0.5 0.8 1 0.8 0.5∏ })({x
∏ })({x
8.0
}8.0,1.0,0{
}})9({,})5({,})2({{
})({)(
}9,5,2{
=
=
=
=
=
∏∏∏
∏∏ ∈
Sup
Sup
xSupA
A
Ax

36. 1: Measurement of Fuzzy Sets1: Measurement of Fuzzy Sets
 Is Defined to Show the Degree and value of FuzzyIs Defined to Show the Degree and value of Fuzzy
Set ambiguity (Set ambiguity (à is finiteà is finite).).
 De Luca & Termini Definition:De Luca & Termini Definition:
)'()(:4
)'()(),2/1
,,2/1(':3
max)(:2
0)(::1
~~~~
~~
'
'
~~
~
~~
~~~
~~~
AdnAdthenAofrycomlementabeAnif
AdAdthenifand
AthanDefinitemorebeAif
valueimizedanduniqueahasAd
AdthenSetDefiniteabeAif
AAA
AAA
=
≤≤≥
≥≤
=
µµµ
µµµ

37. 1: Measurement of Fuzzy Sets1: Measurement of Fuzzy Sets
 Entropy of Fuzzy Sets (De Luca &Entropy of Fuzzy Sets (De Luca &
Termini):Termini):
∑
∑
=
=
=
−−+−=
−=
∈+=
n
i
i
A
i
A
n
i
i
A
xSkAd
xxxxxSfunctionShenon
NumberConsDefinedk
xxkAH
XxAnHAHAd
1
~
1
~
~~~
))(()(
))1ln().1()ln(.()(:
.:
))((ln).()(
),()()(
~
~~
µ
µµ

38. Ex. Of Measurement of Fuzzy SetsEx. Of Measurement of Fuzzy Sets
 Ã is defined as set of numbers near to 10:Ã is defined as set of numbers near to 10:
Ã={(7,0.1),(8,0.5) ,(9,0.8) ,(10,1) ,(11,0.8) ,(12,0.5) ,(13,0.1)}Ã={(7,0.1),(8,0.5) ,(9,0.8) ,(10,1) ,(11,0.8) ,(12,0.5) ,(13,0.1)}
Ĩ= {(6,0.1),(7,0.3),(8,0.4) ,(9,0.7) ,(10,1) ,(11,0.7) ,(12,0.4) ,Ĩ= {(6,0.1),(7,0.3),(8,0.4) ,(9,0.7) ,(10,1) ,(11,0.7) ,(12,0.4) ,
(13,0.3),(14,0.1)}(13,0.3),(14,0.1)}
If k=1 then:If k=1 then:
d (Ã)= 0.325+0.693+0.501+0+0.501+0.693+0.325=3.038d (Ã)= 0.325+0.693+0.501+0+0.501+0.693+0.325=3.038
d ( Ĩ )= 0.325+0.611+0.673+0.501+0+0.501+0.673+0.611+0.325= 4.22d ( Ĩ )= 0.325+0.611+0.673+0.501+0+0.501+0.673+0.611+0.325= 4.22
Ĩ is more fuzzier than à and has moreĨ is more fuzzier than à and has more
Entropy.Entropy.

39. Graph of S FunctionGraph of S Function
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
M(x)
S(x)

40. 2: Measurement of Fuzzy Sets2: Measurement of Fuzzy Sets
 This Definition will care more aboutThis Definition will care more about
the differences that exist between athe differences that exist between a
set and its complementary.set and its complementary.
 If à be fuzzy set and nà be itsIf à be fuzzy set and nà be its
complementary then below definitecomplementary then below definite
definitions has no necessity to be true:definitions has no necessity to be true:
(it means fuzzy sets may have contradiction with(it means fuzzy sets may have contradiction with
definite sets)definite sets)
φ=∩
=∪
~~
~~
AnA
XAnA

41. 2: Measurement of Fuzzy Sets2: Measurement of Fuzzy Sets
 Yager Definition for Ã:Yager Definition for Ã:
∑∑
∑∑
∑
==
==
=






−=





−=
−==
−=−=
−==
−=
=





−=
==
n
i
i
A
n
i
i
An
i
A
i
A
i
An
n
i
i
A
n
i
i
An
i
A
i
A
i
An
p
p
p
pn
i
p
i
An
i
A
p
p
p
xAnADxxAnAD
xxthenpif
xAnADxxAnAD
xxthenpif
ASupp
AnAD
Af
pxxAnAD
ASuppSSnSSDAssume
1
2/12~~
2
1
2/12~~
2
1
~~
1
1
~~
1
/1~
~~
~
/1
1
~~
~
/1
1)(2),()()(),(
)(1)(:2
1)(2),()()(),(
)(1)(:1
)(
),(
1)(
,...3,2,1,)()(),(
:)(),(
~~~
~~
~~~
~~
~~
µµµ
µµ
µµµ
µµ
µµ

42. Ex. Measurement of Fuzzy SetsEx. Measurement of Fuzzy Sets
 Ã is defined as set of numbers near to 10:Ã is defined as set of numbers near to 10:
Ã={(7,0.1),(8,0.5) ,(9,0.8) ,(10,1) ,(11,0.8) ,(12,0.5) ,(13,0.1)}Ã={(7,0.1),(8,0.5) ,(9,0.8) ,(10,1) ,(11,0.8) ,(12,0.5) ,(13,0.1)}
Ĩ= {(6,0.1),(7,0.3),(8,0.4) ,(9,0.7) ,(10,1) ,(11,0.7) ,(12,0.4) ,(13,0.3),(14,0.1)}Ĩ= {(6,0.1),(7,0.3),(8,0.4) ,(9,0.7) ,(10,1) ,(11,0.7) ,(12,0.4) ,(13,0.3),(14,0.1)}
489.0
9
6.4
1)(9)(
6.4),(
457.0
7
8.3
1)(7)(
8.3),(:1
~
1
~
~~
1
~
1
~
~~
1
=−==
=
=−==
==
BfBSupp
BnBD
AfASupp
AnADthenpif
407.0
3
78.1
1)(3)(
78.1),(
347.0
65.2
73.1
1)(65.2)(
73.1),(:2
~
2
2/1~
~~
2
~
2
2/1~
~~
2
=−==
=
=−==
==
BfBSupp
BnBD
AfASupp
AnADthenpif

43. Chapter 3:Chapter 3:
Propagation(Propagation() principle &) principle &
its Applicationsits Applications

44. Propagation(Propagation() principle & Apps.) principle & Apps.
 If X=X1*X2*…*Xr and Ã1, Ã2, Ãr be fuzzyIf X=X1*X2*…*Xr and Ã1, Ã2, Ãr be fuzzy
sets of X1, X2, …, Xr and y= f(x1,…,xr)sets of X1, X2, …, Xr and y= f(x1,…,xr)
then:then:


 ≠
=
∈==
=



 ≠
=
∈……==
−
−
otherwise
yfxSup
y
thenrif
otherwise
yfxxMinSup
y
A
I
I
AA
I
I
r
0
)()}({
)(
X}xf(x),y|(y)){(y,I
:1
0
)()}(),...,({
)(
X}xr),(x1,xr),,f(x1,y|(y)){(y,I
1
~
1
~
~
~
~
~~
1~
~
φµ
µ
µ
φµµ
µ
µ

45. Ex. Propagation(Ex. Propagation() principle) principle
 Ã={(-1,0.5) ,(0,0.8) ,(1,1) ,(2,0.4)}, fÃ={(-1,0.5) ,(0,0.8) ,(1,1) ,(2,0.4)}, f
(x)=x*x(x)=x*x
Ĩ=f (Ã)={(0,0.8) ,(1,1) ,(4,0.4)}Ĩ=f (Ã)={(0,0.8) ,(1,1) ,(4,0.4)}

46. Propagation(Propagation() principle and Second) principle and Second
Types of Fuzzy SetsTypes of Fuzzy Sets
 μμÃÃ={(ui,={(ui, μμuiui (x))| x(x))| xЄЄX, ui,X, ui, μμuiui (x)(x) ЄЄ[0,1]}[0,1]}
 μμĨĨ={(vi,={(vi, μμuiui (x))| x(x))| xЄЄX, vi,X, vi, μμvivi (x)(x) ЄЄ[0,1]}[0,1]}
)]}(),1{[(
)}(),(min{sup)(
]}1,0[,),,min(|))(,{()()(
)}(),(min{sup)(
]}1,0[,),,max(|))(,{()()(
~~
~~
~~~~~~
~~
~~~~~~
),min(
),max(
i
A
i
A
vu
vuwBA
iiii
BABABA
vu
vuwBA
iiii
BABABA
uu
xxw
vuvuwwwxx
xxw
vuvuwwwxx
ii
ii
ii
ii
µµ
µµµ
µµµµ
µµµ
µµµµ
−=
=
∈==∩=
=
∈==∪=
⊄
=∩
∩∩
=∪
∪∪

47. Ex. Propagation(Ex. Propagation() principle and Second) principle and Second
Types of Fuzzy SetsTypes of Fuzzy Sets
 X=1 ,2 ,3 ,… ,10X=1 ,2 ,3 ,… ,10
 Ã=small Natural NumbersÃ=small Natural Numbers
 Ĩ= Natural Numbers near to 4Ĩ= Natural Numbers near to 4
Ui Vi W=max(Ui, Vi) µui(3) µvi(3) Min{µui(3), µvi(3)}
0.8 1 0.8 1 1 1
0.8 0.8 0.8 1 0.5 0.5
0.8 0.7 0.7 1 0.3 0.3
0.7 1 0.7 0.5 1 0.5
0.7 0.8 0.7 0.5 0.5 0.5
0.7 0.7 0.7 0.5 0.3 0.3
0.6 1 0.6 0.4 1 0.4
0.6 0.8 0.6 0.4 0.5 0.4
0.6 0.7 0.6 0.4 0.3 0.3
)}3.0,7.0(),5.0,8.0(),1,1{(}3,...,1|))3(,{()3(
)}4.0,6.0(),5.0,7.0(),1,8.0{(}3,...,1|))3(,{()3(
~
~
===
===
jv
iu
j
i
vj
I
ui
A
µµ
µµ
sup={1,0.5}=1sup={1,0.5}=1
0.8= min {ui, vi}0.8= min {ui, vi}
Sup{0.5,0.5,0.5,0.3}Sup{0.5,0.5,0.5,0.3}
=0.5=0.5
0.7= min {ui, vi}0.7= min {ui, vi}
Sup{0.4,0.4,0.3}=Sup{0.4,0.4,0.3}=
0.40.4
0.6= min {ui, vi}0.6= min {ui, vi}
)}4.0,6.0(),5.0,7.0(),1,8.0{()3(~~ =
∩BA
µ

48. Fuzzy Numbers & AlgebraicFuzzy Numbers & Algebraic
Opr.Opr.
 Only one x ЄR can be exists that has μM=1Only one x ЄR can be exists that has μM=1
 μM in a specific interval is continuous.μM in a specific interval is continuous.
 Almost 5#{(3,0.8),Almost 5#{(3,0.8), (4,1),(5,1)(4,1),(5,1) ,(6,0.7)},(6,0.7)}
 Almost 5={(3,0.2),(4,0.6),(5,1),(6,0.7),(7,0.1)}Almost 5={(3,0.2),(4,0.6),(5,1),(6,0.7),(7,0.1)}
 Almost 10={(8,0.3),(9,0.7),(10,1),(11,0.7),Almost 10={(8,0.3),(9,0.7),(10,1),(11,0.7),
-0.1
0.1
0.3
0.5
0.7
0.9
1.1
0 2 4 6 8 10
Trapezoidal Number
-0.1
0.1
0.3
0.5
0.7
0.9
1.1
0 2 4 6 8 10
Triangular Number

49. Fuzzy Numbers & AlgebraicFuzzy Numbers & Algebraic
Opr.Opr.
 * operators will became true if:* operators will became true if:
For x1>y1 and x2>y2 it will resultFor x1>y1 and x2>y2 it will result
x1*x2>y1*y2x1*x2>y1*y2
Example: f (x, y)= x + yExample: f (x, y)= x + y
Algebraic Operations in Fuzzy Number usageAlgebraic Operations in Fuzzy Number usage
are:are:
ϕ,,,,,, ⊗Θ⊕→÷×−+

50. Fuzzy Numbers & AlgebraicFuzzy Numbers & Algebraic
Opr.Opr.
).()(}|))(,{(..)(
)/1()(}|))(,{(/1)(
)()(}|))(,{()(
:
)()(
)}(),(min{
~~~
~
~1~1~
~
~~~
~
~
1
~
~~~~
..
~
1~
~
)()(
xxXxxxMxxf
xxXxxxMxxf
xxXxxxMxxf
Example
xSupz
xxSup
MMM
M
MMM
M
MMM
M
MzfxMf
NMNM
λµµµλλ
µµµ
µµµ
µµ
µµµ
λλ
=→∈=→=
=→∈=→=
−=→∈=−→−=
=
=
−−
−
−
−−
∈
⊗

51. Fuzzy Numbers & AlgebraicFuzzy Numbers & Algebraic
Opr.Opr.
0)(
0
)()(
)()()(
:
~~
~~
~~~~
~~~~
=Θ⊕
=⊕
⊕
⊕=⊕
Θ⊕Θ=⊕Θ
MM
MM
memberpassivehas
ityAssociativ
MNNM
NMNM
Summation
)()()(
11
)()(
)()(
)()(
:
~~~~~~~
1~~~~
~~~~
~~~~
~~~~
NMPMNPM
MMMM
memberpassivehas
ityAssociativ
MNNM
NMNM
NMNM
tionMultiplica
⊗+⊗=⊕⊗
=⊗=⊗
⊗
⊗=⊗
⊗Θ=⊗Θ
⊗ΘΘ=⊗
−

52. Fuzzy Numbers & AlgebraicFuzzy Numbers & Algebraic
Opr.Opr.
.
)}(),(min{
)}(),(min{
)}(),(min{)(
)()(
:nSubtractio
~~
~~~~
~~
~~
~~~~
NumberfuzzyisNMSo
yxSup
yxSup
yxSupz
NMNM
NMyxz
NMyxz
NMyxzNM
Θ
=
−=
=
Θ⊕=Θ
−+=
+=
−=Θ
µµ
µµ
µµµ
.
)}(),(min{
)}/1(),(min{
)}(),(min{)(
)()(
:Division
~~
.
.
1~~~~
1~~
~~
~~~~
NumberfuzzyisNMSo
yxSup
yxSup
yxSupz
NMNM
NMyxz
NMyxz
NMyxzNM
φ
µµ
µµ
µµµ
φ
φφ
−
=
=
=
−
=
=
=
⊗=

53. Ex. Fuzzy Numbers & AlgebraicEx. Fuzzy Numbers & Algebraic
Opr.Opr.
)}2.0,12(),4.0,9(
),2.0,8(]),4.0,1max[,6(
]),7.0,3.0max[,4(
),3.0,3(),3.0,2{(
)}2.0,4(),1,3(),7.0,2{(
)}4.0,3(),1,2(),3.0,1{(
~~
~
~
=⊗
=
=
NM
N
M
The result is not fuzzy number because it has not ConvexThe result is not fuzzy number because it has not Convex
Trend.Trend.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 3 6 9 12 15
2 * 3 = Almost 6
M(x)

54. Fuzzy Numbers Intervals:Fuzzy Numbers Intervals:
 TriangularTriangular
 TrapezoidalTrapezoidal
 L-R IntervalL-R Interval
FunctionFunction
(Asymmetric)(Asymmetric)
-0.1
0.1
0.3
0.5
0.7
0.9
1.1
0 2 4 6 8 10
Trapezoidal Number
-0.1
0.1
0.3
0.5
0.7
0.9
1.1
0 2 4 6 8 10
Triangular Number
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 5 10 15
L-R Interval Function
M(x)

55. L-R Interval Function (Asymmetric)L-R Interval Function (Asymmetric)
0)1(
01)(
10)(
1)0(
=
>∀<
<∀>
=
L
xforxL
xforxL
L
2
~
)(}1,0max{
)(})1(,0max{)(
:,),,(
,
)(
~
xp
xp
LR
M
exLx
exLxxL
Examplem
Mofvaluemeanism
numberspositiveare
mxfor
mx
R
mxfor
xm
L
x
−
−
=−
=−=






≥




 −
≤




 −
=
βα
βα
β
α
µ

56. Ex. L-R Interval FunctionEx. L-R Interval Function
5,3,2
21
1
)(
1
1
)( 2
===
+
=
+
= m
x
xR
x
xL βα









≥
−
+
=




 −
≤





 −
+
=




 −
=
5
3
)5(
21
1
3
5
5
2
5
1
1
2
5
)(
2
~
xfor
x
x
R
xfor
x
x
L
x
M
µ
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 2 4 6 8 10 12
L-R Interval Function
M(x)

57. L-R Interval FunctionL-R Interval Function
If m is not an definite number can be introduce by anIf m is not an definite number can be introduce by an
interval of [m- ,m+] and can be shown by:interval of [m- ,m+] and can be shown by:
IntervalDefiniteba
NumberDefinitemm
Examplemm
Mofervalmeanismm
numberspositiveare
mxfor
mx
R
mxmfor
mxfor
xm
L
x
LR
LR
LRM
)0,0,,(
)0,0,,(
:,),,,(
int],[
,
1)(
~
~
βα
βα
β
α
µ −
−
−
−
−
−
−
−
−
−











≥








−
≤≤
≤







 −
=

58. L-R Interval FunctionL-R Interval Function
OperationsOperations
LRLRLR
LRLRLR
LRLRLR
LRLR
LRLRLR
LR
LR
mnmnnmnm
mnmnnmnm
nmnm
mm
nmnm
nN
mM
),,/(),,(),,(
),,.(),,(),,(
),,(),,(),,(
),,(),,(
),,(),,(),,(
),,(
),,(
~
~
γβδαδγφβα
δβγαδγβα
δβγαδγβα
βαβα
δβγαδγβα
δγ
βα
−−≈
++≈⊗
++−=Θ
−=−
+++=⊕
=
=

59. Ex. L-R Interval FunctionEx. L-R Interval Function
OperationsOperations
LR
LRLRLR
LR
LR
NM
OONM
NM
x
xRxL
)4.1,7.0,1(
)6.0,2.0,2()2.0,6.0,2()1,1.1,3(
)2.0,6.0,2()8.0,5.0,1(
1
1
)()(
~~
~~~~
~~
2
−=Θ
−=Θ==⊕
==
+
==

60. Ex. L-R Interval FunctionEx. L-R Interval Function
OperationsOperations
LRLRLR
LRLRLR
N
M
M
LRLR
mnmnnmnm
else
x
x
x
R
x
x
L
x
else
x
x
xelse
x
x
xelse
x
x
x
x
R
x
x
L
x
else
z
zRzLNM
)9.0,8.0,6()3.0,1.0,3()1.0,2.0,2(
),,.(),,(),,(
0
3.39.21
3
3.0
3
3
1.0
3
)(
0
1.28.11
)(
20
21
1.0
2
11
20
21
2.0
2
11
2
1.0
2
2
2.0
2
)(
0
111
)()()3.0,1.0,3()1.0,2.0,2(
~
~
~
~~
≈⊗
++≈⊗


 ≤≤






⇒
≥




 −
≤




 −
=


 ≤≤
=









⇒









≥
≥≤




 −
≤−
≤
≤≤




 −
≤−
⇒
≥




 −
≤




 −
=


 ≤≤−
====
δβγαδγβα
µ
µ
µ

61. Chapter 4:Chapter 4:
Functions & FuzzyFunctions & Fuzzy
AnalyzingAnalyzing

62. Functions & Fuzzy AnalyzingFunctions & Fuzzy Analyzing
Fuzzy functions are defined as generalFuzzy functions are defined as general
form of definite functions.form of definite functions.
3 types of fuzzy functions are defined:3 types of fuzzy functions are defined:
fuzzyfuzzy
fuzzydefinite
definitefuzzy
YXf
YXf
YXf
→−
→−
→−
:3
:2
:1

63. 1: Functions & Fuzzy Analyzing1: Functions & Fuzzy Analyzing
valuefuzzyhaveyx
RYXxy
Example
xy
xxfXx
BRangeADomain
YXf
AyfxB
AB
&
,205
:
)(sup)(
)())((,
:
~
1
~
~~
)(
~~
==⊕⊗=
=
≥∈∀
==
→
−
∈
µµ
µµ

64. 2: Functions & Fuzzy Analyzing2: Functions & Fuzzy Analyzing
valuefuzzyhaveyx
xy
Example
YXyxyxy
YofsetfuzzyisYp
SetferenceYX
Rxf
&
)}3.0,27(),7.0,23(),1,20(),8.0,18(),1.0,17{(
)}3.0,6(),7.0,5(),1,4(),8.0,3(),1.0,2{(
:
),(),,()(
)(
Re:&
~~
)(
~
⊕
⊗=
×∈∀= µµ

65. Fuzzy functions ExtremesFuzzy functions Extremes
If f be a function with real numbers on Domain XIf f be a function with real numbers on Domain X
and inf (f) is defined as considered lowest part ofand inf (f) is defined as considered lowest part of
(f) and Sup (f) is defined as considered highest(f) and Sup (f) is defined as considered highest
part of (f) Maximum of (f) can be defined as:part of (f) Maximum of (f) can be defined as:
And is called:And is called: Maximum SetMaximum Set
)inf()sup(
)inf()(
)(~
ff
fxf
x
M −
−
=µ

66. Ex. Fuzzy functions ExtremesEx. Fuzzy functions Extremes
),...}
4
22
,
4
7
(),0,
2
3
(),
2
1
,(
),1,
2
(),
4
22
,
4
(),
2
1
,0{(
2
1
sin
2
1
)1(1
)1()(
)(
sin)(
~
~
~
−
+
=
+=
−−
−−
=
=
=
ππ
π
ππ
µ
M
x
xSin
x
SetMaximumM
xxf
M
-1.5
-1
-0.5
0
0.5
1
1.5
0 1 2 3 4 5 6 7
X
Y
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6 7
X of Maximum Set
M(x)

67. Fuzzy functions ExtremesFuzzy functions Extremes
In second form of fuzzy function definition (f) is defined baseIn second form of fuzzy function definition (f) is defined base
on the definite domain and f (x) is defined base on theon the definite domain and f (x) is defined base on the
fuzzy Range.So its maximum value is not located in itsfuzzy Range.So its maximum value is not located in its
range as a point but also it will berange as a point but also it will be
““maximum set of fuzzy f (x)”maximum set of fuzzy f (x)”..
In order to reach this approach fuzzy maximum andIn order to reach this approach fuzzy maximum and
fuzzy minimum must be defined:fuzzy minimum must be defined:
DxfxfxnD
DxxxfxfMaxM
MMMaxRXMMMin
i
xfnjM
MDx
nfuzzyDefiniten
i
∈=→=
∈==
→
=
∈
)()),((min)(
}|))(),({(sup)(
),...,(),...,(
~
)(,...,1
~~~~
~~
1
~~~
1
~
~~
~
µµ
µ

68. Ex. Fuzzy functions ExtremesEx. Fuzzy functions Extremes
)},(),1,(),,{()()(
)()()()(
))(())((1))((
432
~
311
24
)()(
1
)(
~~~
+−
−−++
+−
==
==
==
αα
µµµ
αααα
αα
xxxMxfxfMax
xfxfMaxxfxfMax
xfxfxf
xfxfxf

69. Integral of Fuzzy FunctionsIntegral of Fuzzy Functions
1.1. Integral of Type 1Integral of Type 1
fuzzy function withfuzzy function with
definite intervaldefinite interval
2.2. Integral of definiteIntegral of definite
function with fuzzyfunction with fuzzy
intervalinterval
3.3. Integral of Type 2Integral of Type 2
fuzzy function withfuzzy function with
definite intervaldefinite interval
4.4. Integral of Type 2Integral of Type 2
fuzzy function withfuzzy function with
fuzzy intervalfuzzy interval

70. 3. Integral of Type 2 fuzzy function3. Integral of Type 2 fuzzy function
with definite intervalwith definite interval
Fuzzy function f (x) is in stage ofFuzzy function f (x) is in stage of αα so:so:
















+=
≥∈→⊆
≥≥≥≥→≥
=≠≠
∈=
∫ ∫
−+
−+
−−++
−+−+
b
a
b
a
xf
dxxfdxxfbaI
xfxfbaxRRba
xfxfxfxfxf
xfxfelsexfxfthenif
y
α
αµ
αα
α
ααµ
αα
αα
αααα
αααα
),)()((
2
1),(
))(&)((],,[,],[
)()()()()('
)()()()(1
]1,0[,)(
~
~~
~
'
~~~
'
~~~~
)(
~

71. Integral of fuzzy function withIntegral of fuzzy function with
definite intervaldefinite interval
 Dubois & Prade show that Fuzzy function f (x) is in stage ofDubois & Prade show that Fuzzy function f (x) is in stage of
αα so:so: ( )
( )
( )LR
LR
LR
b
a
b
a
b
a
LR
Idxxdxxt
dxxdxxsdxxdxxf
x
xR
x
xLxxxxf
Example
dxxtdxxsdxxfbaI
xtxsxfxf
75.3,875.1,21)4,1(750.3
2
)(
875.1
4
)(21)(
21
1
)(
1
1
)(
2
,
4
,)(
:
)(,)(,)(),(
)(),(),()(
~4
1
4
1
4
1
4
1
4
1
2
4
1
2
2
~
~
~
===
====
+
=
+
==








=
=
∫∫
∫∫∫∫
∫ ∫∫

72. Properties of fuzzy IntegralProperties of fuzzy Integral
 If AIf Aαα is defined asis defined as αα stage ofstage of ÃÃ then the Supportthen the Support
Set will become:Set will become:
( ){ }
∫∫
∫
∫∫∫
=







∈∀








=







==



∉
∈
=∈===
∈∈
∈∈
II
I
III
AA
ff
ifonlyandifmeaningfulisf
fffA
functionIntegralfuzzyaasassumedbeAif
Ax
Ax
xAxxxAAAAS
~~
~
~
]1,0[
~
]1,0[
~~
~
]1,0[
~
]1,0[
~
]1,0[
:
:
0
)(|)(,)(
α
αααα
α
α
α
α
ααα
α
α
α
αα
α
µµα αα



73. Properties of fuzzy IntegralProperties of fuzzy Integral
Dubois & Prade show that:Dubois & Prade show that:
)(
:
),()())((
)()(
~~~~
~~
~~~~
~~~
~~~
~
~
∫∫ ∫
∫ ∫∫
∫∫∫
⊕⊆⊕
∈⊕=⊕
⊕=
=−∀−==
∫
∫
II I
b
a
c
b
c
a
f
f
a
b
b
aI
gfgf
thenSetSupportLimitedhadgandfif
Xuugufugf
fff
uuuiffff a
b
a
b
µµ

74. Integral of Definite functions withIntegral of Definite functions with
fuzzy intervalfuzzy interval
Dubois & Prade suggest that:Dubois & Prade suggest that:
)()(
)(sup)(
,)()(
))(),(min(sup)(
:principlenPropagatio
],[
~~
)(:)(
principlenPropagatio
,
00
~
~~
~~
aFbFf
xz
JcdyyfxF
yxz
withthen
baJindefinedisfif
D
axFzxaF
x
c
ba
fz
Jyx
f
F
y
x
F
Θ=
= →
∈=
∫
=
=
∫
∫
∫
=
=
∈
∫
µµ
µµµ

75. Ex. Integral of Definite functionsEx. Integral of Definite functions
with fuzzy intervalwith fuzzy interval
{ }
{ }
)}2.0,8(),8.0,6(,
)1,4(),7.0,2(),4.0,0{()(
22)(
]8,4[],[2)(
)2.0,8(),1,7(),7.0,6(
)4.0,6(),1,5(),8.0,4(
:
~
~
~
~
~~
~
~
∫
∫∫
=
==
=∈=
=
=
D
b
a
b
aD
oo
dxxf
xdxdxxf
baxxf
b
a
Example
(a0 ,b0)(a0 ,b0)
IntegralIntegral
(2dx,a0,b0)(2dx,a0,b0)
MinMin
(Mx(a),Mx(b))(Mx(a),Mx(b))
(4,6) 4 0.7
(4,7) 6 0.8
(4,8) 8 0.2
(5,6) 2 0.7
(5,7) 4 1
(5,8) 6 0.2
(6,6) 0 0.4
(6,7) 2 0.4
(6,8) 4 0.2

76. Integral of Definite functions withIntegral of Definite functions with
fuzzy intervalfuzzy interval
{ }
{ }
{ } ∫∫∫∫
∫ ∫
∫
∫
∫∫∫
∫∫∫
⊕⊆+→=+





 −−−
=+
−−−=
=
=++−=−=
==
+−=−=
⊕⊆+→
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
)3.0,8(),8.0,6(),1,4(),7.0,2(),4.0,0())()((
)3.0,14(),3.0,12(),3.0,10(),7.0,8(),8.0,6(),1,4(
),7.0,2(),7.0,0(),4.0,2(),3.0,4(),3.0,6(
)()(
)7.0,2(),8.0,0(),1.0,2(),3.0,4(),3.0,6()(
)3.0,12(),3.0,10(),1,6(),4.0,4(),7.0,2(),4.0,0()(
2))()((5)(3)(
)}3.0,5(),1,4(),7.0,3{()}4.0,3(),1,2(),8.0,1{(
52)(32)(
:
:,
22
~~
b
a
b
a
b
a
b
a
b
a
b
a
b
a
b
a
b
a
b
a
b
a
b
a
b
a
b
a
b
a
b
a
b
a
gfgfdxxgxf
dxxgdxxf
dxxg
dxxf
xdxxgxfxxdxxgxxdxxf
ba
xxgxxf
Example
gfgfRIgf

77. Integral of Definite functions withIntegral of Definite functions with
fuzzy intervalfuzzy interval
∫∫∫
∫∫∫
⊕⊆⇒===
⊕=+⇒→→ ++
~~~
~
~
~
~
~
~
"'
~~~~~~~~~
),("),,('),,(
:,:,
DDD
b
a
b
a
b
a
gffbcDcaDbaD
gfgfRIgforRIgf

78. Differentiation of Definite functionsDifferentiation of Definite functions
With Fuzzy Domains & RangesWith Fuzzy Domains & Ranges
In this section Differentiation of Definite function must beIn this section Differentiation of Definite function must be
calculated in a convex fuzzy point and with Support Set ofcalculated in a convex fuzzy point and with Support Set of
[a, b] this will cause the obtained answer from f’ (x)[a, b] this will cause the obtained answer from f’ (x)
become a fuzzy answer.become a fuzzy answer.
)}6.0,3(),1,0{()(')}6.0,1(),1,0(),4.0,1{(
3)(')(:.
)(')()()('))(''()()'.(
:thenDescendingn','
)(')()()('))(''()()'.(
)(')('))(''()(')('))(''(
)(sup)()(sup)(
0
~
0
~
23
0
~
0
~
0
~
0
~
0
~
0
~
0
~
0
~
0
~
0
~
0
~
0
~
0
~
0
~
0
~
:thenDescendingorAscending','
0
~
0
~
0
~
)(')('.)('')(')(' 0
~
0
~
0
~
10
~
=→−=
==




⊗⊕



⊗=+=




⊗⊕



⊗⊆+=
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
⊕=+ →⊕⊆+
==
+=+∈ −
XfX
xxfxxfEx
XgXfXgXfXfggfXgf
otandpositivebegfif
XgXfXgXfXfggfXgf
XgXfXgfXgXfXgf
xyxy
notaregfif
XxgxfyxXgfXyfxXf
µµµµ

79. Chapter 5:Chapter 5:
Relations & Fuzzy GraphsRelations & Fuzzy Graphs

80. Fuzzy RelationsFuzzy Relations
Example: If X is most bigger than Y then:Example: If X is most bigger than Y then:
0
0.25
0.5
0.75
1
0 2 4 6 8 10 12 14x/y
M(x,y)
}),(|),(),,{(
,
~
~
YXyxyxyxR
thenRYXif
R
×∈=
⊆
µ






>
−
+
≤
=






≥
<<
−
≤
= yx
xy
yx
yxor
yx
yxy
y
yx
yx
yx
RR
11
)
)(
1
1(
1
0
),(
111
11
10
)(
0
),(
2
~~ µµ
0
0.25
0.5
0.75
1
0 2 4 6 8 10 12 14x/y
M(x,y)

81. Fuzzy RelationsFuzzy Relations
Ex.: X be mostEx.: X be most
bigger than Y:bigger than Y:
Ex.: X be mostEx.: X be most
near to Y:near to Y:
y1 y2 y3 y4
x1 0.8 1 0.1 0.7
x2 0 0.8 0 0
x3 0.9 1 0.7 0.8
y1 y2 y3 y4
x1 0.4 0 0.9 0.6
x2 0.9 0.4 0.5 0.7
x3 0.3 0 0.8 0.5

82. Fuzzy Graphs in Fuzzy SetsFuzzy Graphs in Fuzzy Sets
 If X,Y C R then:If X,Y C R then:
)}(),(min{),(
}|))(,{(
}|))(,{(
~~~
~
~
~
~
yxyx
YyyyB
XxxxA
BAR
B
A
µµµ
µ
µ
≤
∈=
∈=
YXyxyxyxoryx
YXyxyxyxoryx
ZRZR
ZRZR
,),()},(),,(max{),(
,),()},(),,(min{),(
~~~~
~~~~
∈=≥
∈=≤
∪
∩
µµµ
µµµ

83. Ex. Fuzzy Graphs in Fuzzy SetsEx. Fuzzy Graphs in Fuzzy Sets
Ex.: X be mostEx.: X be most
bigger than Y:bigger than Y:
),(~~ yx
ZR∪
µ
Ex.: X be mostEx.: X be most
near to Y:near to Y:
y1 y2 y3 y4
x1 0.8 1 0.1 0.7
x2 0 0.8 0 0
x3 0.9 1 0.7 0.8
y1 y2 y3 y4
x1 0.4 0 0.9 0.6
x2 0.9 0.4 0.5 0.7
x3 0.3 0 0.8 0.5
),(~~ yx
ZR∩
µ
y1 y2 y3 y4
x1 0.8 1 0.9 0.7
x2 0.9 0.8 0.5 0.7
x3 0.9 1 0.8 0.8
y1 y2 y3 y4
x1 0.4 0 0.1 0.6
x2 0 0.4 0 0
x3 0.3 0 0.7 0.5

84. Fuzzy Images in 2-D GraphsFuzzy Images in 2-D Graphs
Ex.:Ex.:
R T = 1R T = 1
}),(|)),({maxmax
}),(|)),(max,{(
}),(|)),(max,{(
}),(|)],(),,{[(
~
~
~
~
)~
])[2(~
])[1(~
~
YXyxyxR
YXyxyxyR
YXyxyxxR
YXyxyxyxR
Ryx
T
Rx
y
Ry
x
R
×∈=
×∈=
×∈=
×∈=
µ
µ
µ
µ
y1 y2 y3 y4 y5 y6
x1 0.1 0.2 0.4 0.8 1.0 0.8
x2 0.2 0.4 0.8 1.0 0.8 0.6
x3 0.4 0.8 1.0 0.8 0.4 0.2
R 1R 1 Max (M (x, y))
x1 1
x2 1
x3 1
y1 y2 y3 y4 y5 y6
R 2R 2 0.4 0.8 1.0 1.0 1.0 0.8

85. Fuzzy Images in n-D GraphsFuzzy Images in n-D Graphs
 If n-D space be considered then Rq can be anIf n-D space be considered then Rq can be an
image in (n-q)-D Space of n-D general space. Soimage in (n-q)-D Space of n-D general space. So
It is obviously that Rq can be a fuzzy set itself.It is obviously that Rq can be a fuzzy set itself.
Rqs can be defined in the direction of independentRqs can be defined in the direction of independent
major axis of n-D spaces Like X-axis or Y-axis inmajor axis of n-D spaces Like X-axis or Y-axis in
Cartesian coordinate or r-axis andCartesian coordinate or r-axis and θθ-axis in-axis in
Cylindrical coordinate.Cylindrical coordinate.
 Ex.: Cylindrical R 2:Ex.: Cylindrical R 2:
y1 y2 y3 y4 y5 y6
x1 0.1 0.2 0.4 0.8 1.0 0.8
x2 0.2 0.4 0.8 1.0 0.8 0.6
x3 0.4 0.8 1.0 0.8 0.4 0.2
R 2 CR 2 C y1 y2 y3 Y4 y5 y6
x1 0.4 0.8 1.0 1.0 1.0 0.8
x2 0.4 0.8 1.0 1.0 1.0 0.8
x3 0.4 0.8 1.0 1.0 1.0 0.8

86. Operations in Fuzzy GraphsOperations in Fuzzy Graphs
(Max-Min. Ops. is Chosen)(Max-Min. Ops. is Chosen)





∈∈∈×=
∈∈∈+×=
⇒




•=
∈∈∈=
∈∈∈=
×∈=×∈=
•
},,|)}],(),({max),,{[(
},,|)}],(),({max
2
1),,{[(
...
*
},,|)}],(*),({max),,{[(
},,|)}}],(),,({min{max),,{[(
}),(|)],(),,{[(}),(|)],(),,{[(
2
~
1
~
2
~
1
~
2
~
1
~
2
~
1
~
~~
1
~
2
~
1
~
2
~
1
~
*
2
~
1
~
2
~
2
~
1
~
ZzYyXxzyyxzxRoR
ZzYyXxzyyxzxRoR
AVE
ZzYyXxzyyxzxRoR
ZzYyXxzyyxzxRoR
ZYzyzyzyRYXyxyxyxR
RRy
RRyAVE
RRy
RRy
RR
µµ
µµ
µµ
µµ
µµ

87. Ex. Operations in Fuzzy Graphs:Ex. Operations in Fuzzy Graphs:
R1 O R2 y1 y2 y3 y4
x1 0.4 0.7 0.3 0.7
x2 0.3 1 0.5 0.8
x3 0.8 0.3 0.7 1
R 2 z1 z2 z3 z4
y1 0.9 0 0.3 0.4
y2 0.2 1 0.8 0
y3 0.8 0 0.7 1
y4 0.4 0.2 0.3 0
y5 0 1 0 0.8
R1 y1 y2 y3 y4 y5
x1 0.1 0.2 0 1 0.7
x2 0.3 0.5 0 0.2 1
x3 0.8 0 1 0.4 0.3
4.0)},(),,{(),(
}0.0,4.0,0.0,2.0,1.0max{),(
0.0}0.0,7.0min{)},(),,(min{
4.0}4.0,0.1min{)},(),,(min{
0.0}8.0,0.0min{)},(),,(min{
2.0}2.0,2.0min{)},(),,(min{
1.0}9.0,1.0min{)},(),,(min{
),(
1111112
~
1
~
11
1551
1441
1331
1221
1111
112
~
1
~
2
~
1
~
2
~
1
~
2
~
1
~
2
~
1
~
2
~
1
~
2
~
1
~
2
~
1
~
==
=
==
==
==
==
==
zxzxzxRoR
zx
zyyx
zyyx
zyyx
zyyx
zyyx
zxRoR
RoR
RoR
RR
RR
RR
RR
RR
µ
µ
µµ
µµ
µµ
µµ
µµ

88. Ex. Operations in Fuzzy Graphs:Ex. Operations in Fuzzy Graphs:
R1 O R2 y1 y2 y3 y4
x1 0.4 0.7 0.3 0.56
x2 0.27 1 0.4 0.8
x3 0.8 0.3 0.7 1
R 2 z1 z2 z3 z4
y1 0.9 0 0.3 0.4
y2 0.2 1 0.8 0
y3 0.8 0 0.7 1
y4 0.4 0.2 0.3 0
y5 0 1 0 0.8
R1 y1 y2 y3 y4 y5
x1 0.1 0.2 0 1 0.7
x2 0.3 0.5 0 0.2 1
x3 0.8 0 1 0.4 0.3
4.0)},(),,{(),(
}0.0,4.0,0.0,04.0,09.0max{),(
0.00.07.0),(),(
4.04.00.1),(),(
0.08.00.0),(),(
04.02.02.0),(),(
09.09.01.0),(),(
),(
1111112
~
1
~
11
1551
1441
1331
1221
1111
112
~
1
~
2
~
1
~
2
~
1
~
2
~
1
~
2
~
1
~
2
~
1
~
2
~
1
~
2
~
1
~
==
=
=×=×
=×=×
=×=×
=×=×
=×=×
•
•
•
•
zxzxzxRoR
zx
zyyx
zyyx
zyyx
zyyx
zyyx
zxRoR
RoR
RoR
RR
RR
RR
RR
RR
µ
µ
µµ
µµ
µµ
µµ
µµ

89. Ex. Operations in Fuzzy Graphs:Ex. Operations in Fuzzy Graphs:
R1 O R2 y1 y2 y3 y4
x1 0.7 0.85 0.65 0.75
x2 0.6 1 0.65 0.9
x3 0.9 0.65 0.85 1
R 2 z1 z2 z3 z4
y1 0.9 0 0.3 0.4
y2 0.2 1 0.8 0
y3 0.8 0 0.7 1
y4 0.4 0.2 0.3 0
y5 0 1 0 0.8
R1 y1 y2 y3 y4 y5
x1 0.1 0.2 0 1 0.7
x2 0.3 0.5 0 0.2 1
x3 0.8 0 1 0.4 0.3
[ ]
[ ]
[ ]
[ ]
[ ]
7.0)},(),,{(),(
}35.0,7.0,4.0,2.0,5.0max{),(
35.00.07.0
2
1),(),(
2
1
7.04.00.1
2
1),(),(
2
1
4.08.00.0
2
1),(),(
2
1
2.02.02.0
2
1),(),(
2
1
5.09.01.0
2
1),(),(
2
1
),(
1111112
~
1
~
11
1551
1441
1331
1221
1111
112
~
1
~
2
~
1
~
2
~
1
~
2
~
1
~
2
~
1
~
2
~
1
~
2
~
1
~
2
~
1
~
==
=
=+=



 +
=+=



 +
=+=



 +
=+=



 +
=+=



 +
zxzxzxRoR
zx
zyyx
zyyx
zyyx
zyyx
zyyx
zxRoR
RoRAVE
RoR
RR
RR
RR
RR
RR
AVE
AVE
AVE
µ
µ
µµ
µµ
µµ
µµ
µµ

90. Answer Comparison:Answer Comparison:

91. Properties ofProperties of Max-MinMax-Min CombinationCombination
 Max-Min function is Commutative:Max-Min function is Commutative:
3
1
~
1
~
1
~
1
~
1
~
2
~
3
~
1
~
2
~
3
~
)()()( RRoRoRRoRoRRoRoR ==
:Pr:
,
),(),(
),(),(
:Pr
),(:Pr
1),(:Pr
~
~
~
~
~~
~~
~
~
opertyMirrorhasRExample
Xyx
xyxx
yxxx
IfopertyMirrorweakhasR
XxxxIfopertyMirrorhasR
XxxxIfopertyMirrorhasR
RR
RR
R
R
∈∀




≥
≥
−
∈∀≥−
∈∀=
µµ
µµ
εµε
µ
R y1 y2 y3 y4
x1 1 0.85 0.65 0.75
x2 0.6 1 0.65 0.9
x3 0.9 0.65 1 1
x4 0.9 0.5 0.6 1

92. Properties ofProperties of Max-MinMax-Min CombinationCombination
XyxyxyxRRoRifopertiesTaadihasR
xyyxifopertiesSymmetricnonhasR
Xyxxyyx
ifopertiesSymmetricAntiCompletelyhasR
XyxxyyxifopertiesSymmetricAntihasR
XyxxyyxifopertiesSymmetrichasR
RRoR
RR
RR
RR
RR
∈∀≤⊆
≠−
∈∀=>
−
∈∀≠−
∈∀=
,),(),(:Pr
),(),(:Pr
,0),(0),(
:Pr
,),(),(:Pr
,),(),(:Pr
~~~
~~
~~
~~
~~
~~~~
~
~
~
~
µµ
µµ
µµ
µµ
µµ

93. Ex. Properties ofEx. Properties of Max-MinMax-Min
CombinationCombination
Symmetric SymmetricAnti −
SymmetricNon − Taadi
R y1 y2 y3 y4
x1 1 0.9 0.7 0.8
x2 0.9 1 0.7 0.9
x3 0.7 0.7 1 1
x4 0.8 0.9 1 1
R y1 y2 y3 y4
x1 1 0.9 0.7 0.8
x2 0.6 1 0.7 0.9
x3 0.9 0.7 1 1
x4 0.9 0.5 0.6 1
R y1 y2 y3 y4
x1 1 0 0.7 0
x2 0.6 1 0 0.9
x3 0 0.7 1 1
x4 0.9 0 0 1
R y1 y2 y3 y4
x1 1 0.9 0.7 0.8
x2 0.6 1 0.7 0.9
x3 0.9 0.7 1 1
x4 0.9 0.9 1 1
R y1 y2 y3 y4
x1 0.2 1 0.4 0.4
x2 0 0.6 0.3 0
x3 0 1 0.3 0
x4 0.1 1 1 0.1
RoR y1 y2 y3 y4
x1 0.2 0.6 0.4 0.2
x2 0 0.6 0.3 0
x3 0 0.6 0.3 0
x4 0.1 1 0.3 0.1
SymmetricAntiCompletly −

94. Properties ofProperties of Max-MinMax-Min CombinationCombination
.&.5
.
:&.4
.&.3
:.2
:.1
~~~~
1
~
2
~
2
~
1
~
2
~
1
~
2
~
1
~
2
~
1
~
~~~~
2
~
1
~
2
~
1
~
2
~
1
~
1
~
SymmetricbecomewillRRoRthenSymmetricbeRif
SymmetricbecomewillRoRRoR
thenSymmetricbeRRif
mirroricbecomewillRoRthenmirroricbeRRif
RoRRthenmirroricbeRif
RoRRRoRRthenmirroricbeRif
n
=
⊆
⊆⊆

95. Properties ofProperties of Max-MinMax-Min CombinationCombination
.&&
:&.8
:&.7
),(),(,
:&.6
~
2
~
1
~
1
~
2
~
2
~
1
~
2
~
1
~~~
~
~
~~
TaadicbecomewillRoRRRRoR
thenTaadicbeRRif
RRoR
thenTaadicMirroricbeRif
xxyxXyx
thenTaadicSymmetricbeRif
RR
=
=
≤∈∀ µµ

96. Properties Validity:Properties Validity:
Max-Min Max-Mul Max-Mean Op. with Associativity
1 TRUE - TRUE Arbitrary
2 TRUE TRUE - Arbitrary
3 TRUE - TRUE Arbitrary
4 TRUE TRUE - Arbitrary
5 TRUE TRUE - TRUE
6 TRUE - - Arbitrary
7 TRUE TRUE TRUE Arbitrary
8 TRUE TRUE TRUE Arbitrary

97. Fuzzy GraphsFuzzy Graphs










=






×∈



=
]0.1),,[(],7.0),,[(],7.0),,[(
]3.0),,[(],5.0),,[(],0.0),,[(
]3.0),,[(],3.0),,[(],0.0),,[(
),(
:
),(),(),,(),(
~
~
~
ccbcac
cbbbab
cabaaa
ji
jiji
G
jiji
xxxxxx
xxxxxx
xxxxxx
xxG
Example
EExxxxxxxxG µ

98. Fuzzy GraphsFuzzy Graphs










=










=
×∈∀≤
]0.5),,[(],7.0),,[(],2.0),,[(
]3.0),,[(],3.0),,[(],0.0),,[(
]3.0),,[(],1.0),,[(],0.0),,[(
),(
]0.1),,[(],7.0),,[(],7.0),,[(
]3.0),,[(],5.0),,[(],0.0),,[(
]3.0),,[(],3.0),,[(],0.0),,[(
),(
:
),(),(),(
:.),(),(
~
~
~~
~~
ccbcac
cbbbab
cabaaa
ji
ccbcac
cbbbab
cabaaa
ji
jiji
G
ji
H
jiji
xxxxxx
xxxxxx
xxxxxx
xxH
xxxxxx
xxxxxx
xxxxxx
xxG
Example
EExxxxxx
DimSamewithifxxGofSpanbecomewillxxH
µµ

99. Fuzzy GraphsFuzzy Graphs
 The Length & distance between 2The Length & distance between 2
nodes are defined as follows:nodes are defined as follows:
)}({min)(
),(
1
)(
1 1
PLPD
xx
PL
i
i
n
i ii
=
= ∑= +µ

100. Fuzzy Forests:Fuzzy Forests:
Forest are that graphs that have no loopForest are that graphs that have no loop