18. 2. Fuzzy Relations Objectives: Crisp and Fuzzy Relations
Projections and Cylindrical Extensions*
Extension Principle
Compositions of Fuzzy Relations
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19. Cartesian Product: crisp
Let A and B be two crisp subsets in X and Y, respectively. The Cartesian product of A and B, denoted by AB, is defined by
Let X={0, 1}, Y={a,b,c}. If A=X and B=Y, then
AB={(0,a), (0,b), (0,c), (1,a), (1,b), (1,c)}
BA={(a,0), (b,0), (c,0), (a,1), (b,1), (c,1)}
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20. Fuzzy Relationships
A fuzzy relationship over the pair X, Y is defined as a fuzzy subset of the Cartesian product XY.
If X={0, 1}, Y={a,b,c}, then
A = {0.1/(0,a), 0.6/(0,b), 0.8/(0,c),
0.3/(1,a), 0.5/(1,b), 0.7/(1,c)}
is a fuzzy relationship over the space XY.
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21. Cartesian Product: fuzzy
Let A and B be fuzzy sets in X and Y, respectively. The Cartesian product of A and B, denoted by AB, is a fuzzy set in the product space XY with the membership function:
Assume X={0,1} and Y={a,b,c}
Let A=1.0/ /1, B=0.2/a + 0.5/b+ 0.8/c.
Then AB is a fuzzy relationship over XY.
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22. Cylindrical Extension*
Assume X and Y are two crisp sets and let A be a fuzzy subset of X. The cylindrical extension of A to XY, denoted by , is a fuzzy relationship on XY.
Assume X={a,b,c} and Y={1,2}.
Let A={1/a, 0.6/b, 0.3/c}. Then the cylindrical extension of A to XY is {1/(a,1), 1/(a,2), 0.6/(b,1), 0.6/(b,2), 0.3/(c,1), 0.3/(c,2)}
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23. Cylindrical Extension*
A(x) x y x
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24. Projection*
Assume A is a fuzzy relationship on XY. The projection of A onto X is a fuzzy subset A of X, denoted by A=Projx A,
Assume X = {a,b,c} and Y = {1,2}. Let A={1/(a,1), 0.6/(a,2), 0.8/(b,1), 0.6/(b,2), 0.3/(c,1), 0.5/(c,2)}. Then Projx A = {1/a, 0.6/b, 0.5/c}.
Projy A = {0.8/1, 0.6/2}.
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25. Projection*
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26. Extension Principle
Assume X and Y are two crisp sets and let f be a mapping form X into Y, f: XY, such that xX, f(x) = y Y. Assume A is a fuzzy subset of X, using the extension principle, we can define f(A) as a fuzzy subset of Y such that
Denote B = f(A), then B is a fuzzy subset of Y such that for each y Y
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27. Example 2-3 Assume X = {1, 2, 3} and Y = {a, b, c, d, e}.
Let f be defined by f(1) = a, f(2) = e, f(3) = b.
Let A = {1.0/1, 0.3/2, 0.7/3} be a fuzzy subset,
then B = f(A) = {1.0/a, 0.3/e, 0.7/b}.
Let A = 0.1/ / / / /2
and f(x) = x2 3.
Then B = 0.1/ / / / /1
= 0.8/3 + (0.40.9)/2 + (0.10.3)/1
= 0.8/ / /1
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28. Binary Fuzzy Relations
Let X and Y be two universes of discourse. Then
is a binary fuzzy relation in XY.
Examples of binary fuzzy relation:
y is much greater than x. (x and y are numbers)
x is close to y. (x and y are numbers)
x depends on y. (x and y are events)
x and y look alike. (x and y are persons, objects, etc.)
If x is large, then y is small. (x is an observed reading and y is a corresponding action)
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29. Max-min Composition
Let R1 and R2 be two fuzzy relations defined on XY and YZ, respectively. The max-min composition of R1 and R2 is a fuzzy set defined by
Max-min product: the calculation of is almost the same as matrix multiplication, except that  and  are replaced by  and , respectively.
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30. Max-product Composition
Let R1 and R2 be two fuzzy relations defined on XY and YZ, respectively. The max-product composition of R1 and R2 is a fuzzy set defined by
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31. Example 2-3 R1 = “x is relevant to y”, R2 = “y is relevant to z”,
X = {1,2,3}, Y={,,,} and Z={a,b}.
Max-min composition:
Max-product composition:
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