CHAPTER 2
FUNDAMENTALS OF FUZZY LOGIC SYSTEMS
Universe X (Element x)
Fuzzy Boundary
Fuzzy Set A
Figure 2.1. Venn diagram of a fuzzy set.
x
Membership Grade µA(x)
1.0
0
Fuzzy Fuzzy
Figure 2.2. The membership function of a fuzzy set.
0
1
A
A’
Figure 2.3. Fuzzy-set complement or fuzzy-logic NOT.
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0 10 20 30 40
0.5
1.0
Membership Grade
Hot
Not Hot
Temperature (°C)
Figure 2.4. An example of fuzzy-logic NOT.
0
1 A
B
A∪B
Figure 2.5. Fuzzy-set union or fuzzy-logic OR.
0
1 A
B
A∩B
Figure 2.6. Fuzzy-set intersection or fuzzy-logic AND.
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MembershipGrade
1
00.0 1.0 2.0 Speed (m/s)
MembershipGrade
1
00.0 10.0 20.0 Power (hp)
MembershipGrade
Power
Speed 1
0
0.0
0.0 20.0
2.0
(a)
(b)
(c)
Figure 2.7. (a) Required speed; (b) Required power; (c) Required speed and power.
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0
1
A
A’
A∪A’
Figure 2.8. An example of excluded middle in fuzzy sets.
0
1 A
B
A⊂ B
B⊂ A
Figure 2.9. An example of grade of inclusion.
µA(x) µB(y)
1.0 1.0
1.0 1.000 0.30.5 0.70.3 0.5 0.7 x∈ [0, 1] y∈ [0, 1]
Figure 2.10. Membership functions of fuzzy sets A and B.
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Figure 2.11. A graphical representation of various fuzzy implications operations.
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0
1
( )A xµ
α x
0α =
( )A xα µ=
1α =0Aα
µ =
1Aαµ =
Figure 2.12. Graphical proof of the representation theorem.
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Figure 2.13. An interpretation of fuzzy resolution
(a) High fuzzy resolution (b) Low fuzzy resolution.
0
1
More Fuzzy
0.5 Less Fuzzy
Figure 2.14. Illustration of fuzziness.
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1.0
0
0.5
1.0
0
0.5
1.0
0
0.5
(a) (b)
(c)
( )
1A
A
xµµ−
S S
S
( )
21
y
A
A
xµµ−
( )1
A
A
xµµ−
Figure 2.15. Illustration of three measures of fuzziness:
(a) Closeness to grade 0.5 (b) Distance from ½ cut (c) Inverse of distance from the complement.
1.0
0.5
1.50.5 1.0x
µA1/2
µA
x
µB1.0
3.02.01.0
µA1.0
1.50.5 1.0x
1.0
1.50.5 1.0x
µS
1.25] [0.75,}5.0)(|{21 =≥= xxA Aµ
(i)(ii)
(iii) (iv)
1.0
0.5
1.50.5 1.0x
µA1/2
µA
x
µB1.0
3.02.01.0
µA1.0
1.50.5 1.0x
1.0
1.50.5 1.0x
µS
1.25] [0.75,}5.0)(|{21 =≥= xxA Aµ
(i)(ii)
(iii) (iv)
Figure 2.16. (i) µA, (ii) Support set of A, (iii) A1/2, (iv) µB
1.0
1.50.5 1.0x
µA1/2
µA
x
µB
1.0
3.02.01.0
µB1/2
0.50.5
Figure 2.17. Fuzziness of A and B.
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Figure 2.18. Relation R in a two-dimensional space (plane):
(a) A crisp relation; (b) A fuzzy relation.
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Figure 2.19. Cartesian product ( A A1 × 2 ) relation of:
(a) Two crisp sets; (b) Two fuzzy sets.
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Figure 2.20. A crisp mapping from a product space to a line:
(a) An example of crisp sets (b) An example of fuzzy sets (Extension principle).
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Figure 2.21(a). A non-fuzzy algebraic system with a fuzzy input.
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Figure 2.21(b). Application of the extension principle.
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(a)
Outputy
Input u
(b)
Outputy
Input u
Footprint of
(a)
Outputy
Input u
(b)
Outputy
Input u
Footprint of
Figure 2.22. (a) Fuzzy decision making using a crisp relation (Extension
principle). (b) Fuzzy decision making using a fuzzy relation (Composition).
Membership
Grade µ
x
y
0
MembershipGrade µ
x
y
0
Figure 2.23. An example of fuzzy projection.
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Figure 2.24. (a) A fuzzy relation (Fuzzy set); (b) Its cylindrical extension.
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0
1.0
µ A1(xi)
Figure 2.
0
1.0
µ A(xi)
xi
1 2 3 4
xi
0 1 2 3 4
1.0
25. Discrete membership functions of various fuzzy sets and fuzzy relations.
xi
1 2 3 4
yj
0 1 2 3 4
1.0
µ A2(xi) µB(yj)
Figure 2.25. (Cont’d).
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xi
0 1 2 3 4
1.0
µ A1 ∪ A2(xi)
xi
0 1 2 3 4
1.0
µ A1 ∩ A2(xi)
Figure 2.25. (Cont’d).
µ A →B (xi, yj)
yjxi
Figure 2.25. (Cont’d).
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µ C(A) (xi, yj)
yj xi
Figure 2.25. (Cont’d).
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µ C(B) (xi, yj)
xi yj
Figure 2.25. (Cont’d).
R (xi,yj) : x2=2 0.6
x3=3 0.3
x4=4 0.0
y0=0
x1=1 0.1
x0=0 0.0
0.7
0.4
0.1
y1=1
0.5
0.4
1.0
0.9
0.5
y2=2
0.8
0.7
0.5
0.7
0.3
y3=3
0.4
0.3
0.2
0.4
0.1
y4=4
0.1
0.0
01
23
45
67
8
zk
z = x+y
µC (zk)
1.0
Figure 2.26. Crisp mapping of R(xi, yj) in X x Y to C(zk) in Z.
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