931102 fuzzy set theory chap07.ppt 1

Fuzzy relations

Fuzzy sets defined on universal sets which are Cartesian products.

<x, y>

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Example (information retrieval)

DocumentsD

Key termsT

Binary fuzzy relationR

Membership degree R(d,t): the degree of relevance of document d and the key term t

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Example (equality relation)

)10(max)(c

y||x-, x,yE

y is approximately equal to x

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Representations of fuzzy relation

• List of ordered pairs with their membership grades• Matrices• Mappings• Directed graphs

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Matrices

Fuzzy relation R on XY

X={x1, x2 , …, xn}, Y={y1, y2 , …, ym}

rij=R(xi , yj) is the membership degree of pair (xi , yj)

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Example (very far from)

Example: X={Beijing, Chicago, London, Moscow, New York, Paris, Sydney, Tokyo}

Very far fromVery far from

X

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Mappings

The visual representations

On finite Cartesian products

Document D = {d1, d2 , …, d5}

Key terms T = {t1, t2 , t3, t4 }

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Directed graphs

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Inverse operation

• Given fuzzy binary relation RXY– Inverse R-1YX

– R-1(y,x) = R(x,y)

– (R-1) -1 = RThe transpose of R

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The composition of fuzzy relations P and Q

• Given fuzzy relations PXY, QYZ– Composition on P and Q = P Q = R XZ

– The membership degree of a chain <x, y, z> is determined by the degree of the weaker of the two links, <x, y> and <y, z> .

– R(x, z) = (P Q )(x, z) = max yY min [ P(x, y ), Q(y, z )]

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Example (composition of fuzzy relations)

X = {a,b,c}

Y = {1,2,3,4}

Z = {A,B,C}

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Example (matrix composition)

P XY Q YZ R XZ

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Fuzzy equivalence relations and compatibility relations

• Equivalence: Any fuzzy relation R that satisfies reflexive, symmetric and transitive properties.

– R is Reflexive R(x, x) = 1 for all x X.

– R is symmetric R(x, y) = R(y, x) for all x, y X.

– R is transitive R(x, z) max yY min [ R(x, y ), R(y, z )] all x, z X.

• Compatibility: satisfy reflexive and symmetric

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Example (fuzzy equivalence)

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Example(fuzzy compatibility)

R(1,4)<max {min[R(1,2),R(2,4)], min[R(1,5),R(5,4)]}

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Transitive closure

• The transitive closure of R is the smallest fuzzy relation that is transitive and contains R.

• interactive algorithm to find transitive closure RT

of R 1. Compute R’ = R(R R)

2. If R’R, rename R’ as R and go to step 1. Otherwise, R’= RT.

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Evaluate the algorithm

equal

Firstiteration

seconditeration

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Example (transitive closure)

Transitive closure of R

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Fuzzy Partial ordering• Fuzzy relations that satisfy reflexive, transitive

and anti-symmetric, denoted by xy

– R on X is anti-symmetric R(x,y)>0 and R(y, x)>0 imply that x=y for any x,yX.

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Example (fuzzy partial ordering)

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Projections example

S ={s1, s2, s3}

D ={d1, d2, d3 , d4 , d5}

Projection on S

Projection on D

),( (s)Q maxDd

1 dsQ

),( (s)Q maxSs

2 dsQ

symptoms

diseases

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Projections

Given an n-dimensional fuzzy relation R on X =X1 X2 … Xn and any subset P(any chosen dimensions) of X, the projection RP of R on P for each p P is

Where

),( )(R maxP

P ppRpp

P is the remaining dimensions

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Cylindric extension

Given an n-dimensional fuzzy relation R on X =X1 X2 … Xn , any (n+k)-dimensional relation whose projection into the n dimensions of R yields R is called an extension of R.

An extension of R with respect to Y =Y1 Y2 …

Yh is call the cylindric extension EYR of R into YEYR(x,y) = R(x)

For all x X and y Y.

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Example (Cylindric extension)

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Example of cylindric closure

• cylindric closure : the intersection of the cylindric extensions of R is the original relation R.