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.

2

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).

21

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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