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Fuzzy Set and Opertion

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Outline Fuzzy Set and Crisp Set Expanding concepts Standard operation of fuzzy set Fuzzy relations Operations on fuzzy relations

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Set

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Crisp set and Fuzzy set Crisp set – membership function – membership degree: {0,1} Fuzzy set – membership function: user specify – membership degree: [0,1] 4

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Crisp set and Fuzzy set

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Expression of fuzzy set 6

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Example of fuzzy set 7

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Support Support of A example 8

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Height The maximum value of the membership degree 9

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Normalized fuzzy set – height is 1 – young, adult, and senior are normalized fuzzy sets 10

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-cut -cut set Example If, 11 5

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-cut 12

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Level set Example 13

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Convex fuzzy set 14

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Non-convex fuzzy set 15

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Relation of fuzzy sets A and B are equivalent A is a subset of B A is a proper subset of B 16

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Subset of fuzzy set 17

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Standard operation of fuzzy set Complement Union Intersection 18

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Example Complement 19

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Union Example 20

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Intersection Example 21

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Other operations Disjunctive sum (exclusive OR)

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Other operations

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Other operations

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25 Product set

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26 Product set

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A={a1,a2} B={b1,b2} C={c1,c2} AxBxC = {(a1,b1,c1),(a1,b1,c2),(a1,b2,c1),(a1,b2,c2),(a 2,b1,c1),(a2,b1,c2),(a2,b2,c1), (a2,b2,c2)} 27

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28 Crisp relation A relation among crisp sets is a subset of the Cartesian product. It is denoted by. Using the membership function defines the crisp relation R :

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29 Fuzzy relation A fuzzy relation is a fuzzy set defined on the Cartesian product of crisp sets A 1, A 2,..., A n where tuples (x 1, x 2,..., x n ) may have varying degrees of membership within the relation. The membership grade indicates the strength of the relation present between the elements of the tuple.

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30 Representation methods Matrix (Crisp)(Fuzzy)

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31 Representation methods Digraph (Crisp)(Fuzzy)

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32 α-cut of fuzzy relation Example

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33 α-cut of fuzzy relation

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34 Operations on fuzzy matrices Sum: Example

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35 Operations on fuzzy matrices Max product: C = A ・ B=AB= Example

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36 Max product Example

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37 Max product Example

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38 Max product Example

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39 Operations on fuzzy matrices Scalar product: Example

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40 Operations on fuzzy relations Union relation For n relations

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41 Union relation Example

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42 Operations on fuzzy relations Intersection relation For n relations

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43 Intersection relation Example

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44 Operations on fuzzy relations Complement relation: Example

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45 Composition of fuzzy relations Max-min composition Example

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46 Composition of fuzzy relations

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47 Composition of fuzzy relations Example

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48 Composition of fuzzy relations Example

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49 Composition of fuzzy relations

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