geometry of fuzzy sets
DESCRIPTION
Geometry of Fuzzy Sets. Sets as points. Geometry of fuzzy sets includes Domain X ={ x 1 ,…,x 2 } Range of mappings [0,1] A :X [0,1]. Classic Power Set. Classic Power Set: the set of all subsets of a classic set. Let X ={ x 1 , x 2 , x 3 } Power Set is represented by 2 | X | - PowerPoint PPT PresentationTRANSCRIPT
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Geometry of Fuzzy SetsGeometry of Fuzzy Sets
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@2004Adriano Cruz NCE e IM - UFRJ Geometry of Fuzzy Sets 2
Sets as pointsSets as points Geometry of fuzzy sets includes
Domain X={x1,…,x2}
Range of mappings [0,1]
A:X[0,1]
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@2004Adriano Cruz NCE e IM - UFRJ Geometry of Fuzzy Sets 3
Classic Power SetClassic Power Set Classic Power Set: the set of all
subsets of a classic set.
Let X={x1,x2 ,x3}
Power Set is represented by 2|X|
2|X|={, {x1}, {x2}, {x3}, {x1,x2}, {x1,x3}, {x2,x3}, X}
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@2004Adriano Cruz NCE e IM - UFRJ Geometry of Fuzzy Sets 4
VerticesVertices The 8 sets correspond to 8 bit vectors
2|X|={, {x1}, {x2}, {x3}, {x1,x2}, {x1,x3}, {x2,x3}, X}
2|X|={(0,0,0),(1,0,0),(0,1,0),(0,0,1),(1,1,0),(1,0,1),(0,1,1),(1,1,1)}
The 8 sets are the vertices of a cube
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@2004Adriano Cruz NCE e IM - UFRJ Geometry of Fuzzy Sets 5
The vertices in spaceThe vertices in space
x1
x2
x3
(0,0,0)
(1,1,1)
(1,0,1)
(1,0,0)
(1,1,0)
(0,1,0)
(0,1,1)
(0,0,1)
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@2004Adriano Cruz NCE e IM - UFRJ Geometry of Fuzzy Sets 6
Fuzzy Power SetFuzzy Power Set The Fuzzy Power set is the set of all
fuzzy subsets of X={x1,x2 ,x3} It is represented by F(2|X|) A Fuzzy subset of X is a point in a cube The Fuzzy Power set is the unit
hypercube
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@2004Adriano Cruz NCE e IM - UFRJ Geometry of Fuzzy Sets 7
The Fuzzy CubeThe Fuzzy Cube
x1
x2
x3
(0,0,0)
(1,1,1)
(1,0,1)
(1,0,0)
(1,1,0) (0,1,0)
(0,1,1)
(0,0,1)
A={(x1,0.5),(x2,0.3),(x3,0.7)}
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@2004Adriano Cruz NCE e IM - UFRJ Geometry of Fuzzy Sets 8
Fuzzy OperationsFuzzy Operations Let X={x1,x2} and A={(x1,1/3),(x2,3/4)}
Let A´ represent the complement of A
A´={(x1,2/3),(x2,1/4)}
AA´={(x1,2/3),(x2,3/4)}
AA´={(x1,1/3),(x2,1/4)}
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@2004Adriano Cruz NCE e IM - UFRJ Geometry of Fuzzy Sets 9
Fuzzy Operations in the SpaceFuzzy Operations in the Space
(0,1)
1/4
3/4
(1,1)
(1,0)
x1
x2
1/3 2/3
A
A´
AA´
AA´
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@2004Adriano Cruz NCE e IM - UFRJ Geometry of Fuzzy Sets 10
Paradox at the MidpointParadox at the Midpoint Classical logic forbids the middle point
by the non-contradiction and excluded middle axioms
The Liar from Crete Let S be he is a liar, let not-S be he is
not a liar Since Snot-S and not-SS t(S)=t(not-S)=1-t(S) t(S)=0.5
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@2004Adriano Cruz NCE e IM - UFRJ Geometry of Fuzzy Sets 11
Cardinality of a Fuzzy SetCardinality of a Fuzzy Set The cardinality of a fuzzy set is equal to
the sum of the membership degrees of all elements.
The cardinality is represented by |A|
n
iiA xA
1
)(||
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@2004Adriano Cruz NCE e IM - UFRJ Geometry of Fuzzy Sets 12
DistanceDistance The distance dp between two sets
represented by points in the space is defined as
If p=2 the distance is the Euclidean distance, if p=1 the distance it is the Hamming distance
p
n
i
piBiA
p xxBAd
1
|)()(|),(
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@2004Adriano Cruz NCE e IM - UFRJ Geometry of Fuzzy Sets 13
Distance and CardinalityDistance and Cardinality If the point B is the empty set (the
origin)
So the cardinality of a fuzzy set is the Hamming distance to the origin
n
iiA
n
iiA
xAAd
xAd
1
1
1
1
)(||),(
0)(),(
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@2004Adriano Cruz NCE e IM - UFRJ Geometry of Fuzzy Sets 14
Fuzzy CardinalityFuzzy Cardinality
(0,1)
3/4
(1,1)
(1,0)
x1
x2
1/3
A
|A|=d1(A,)
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@2004Adriano Cruz NCE e IM - UFRJ Geometry of Fuzzy Sets 15
Fuzzy EntropyFuzzy Entropy How fuzzy is a fuzzy set?
Fuzzy entropy varies from 0 to 1.
Cube vertices has entropy 0.
The middle point has entropy 1.
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@2004Adriano Cruz NCE e IM - UFRJ Geometry of Fuzzy Sets 16
Fuzzy Entropy GeometryFuzzy Entropy Geometry
(0,1)
3/4
(1,1)
(1,0)
x1
x2
1/3
A
a
b
),(),(
)( 1
1
far
near
AAdAAd
ba
AE
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@2004Adriano Cruz NCE e IM - UFRJ Geometry of Fuzzy Sets 17
Fuzzy Operations in the SpaceFuzzy Operations in the Space
(0,1)
1/4
3/4
(1,1)
(1,0)
x1
x2
1/3 2/3
A
A´
AA´
AA´
´
´)(
AA
AAAE
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@2004Adriano Cruz NCE e IM - UFRJ Geometry of Fuzzy Sets 18
Fuzzy entropy, max and minFuzzy entropy, max and min T(x,y) min(x,y) max(x,y)S(x,y) So the value of 1 for the middle point
does not hold when other T-norm is chosen.
Let A= {(x1,0.5),(x2,0.5)} E(A)=0.5/0.5=1 Let T(x,y)=x.y and C(x,y)=x+y-xy E(A)=0.25/0.75=0.333…
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@2004Adriano Cruz NCE e IM - UFRJ Geometry of Fuzzy Sets 19
SubsetsSubsets Sets contain subsets.
A is a subset of B (AB) iff every element of A is an element of B.
A is a subset of B iff A belongs to the power set of B (AB iff A2B).
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@2004Adriano Cruz NCE e IM - UFRJ Geometry of Fuzzy Sets 20
Subsets and implicationSubsets and implication Subsethood is equivalent to the
implication relation. Consider two propositions P and Q. A is a subset of B iff there is no element
of A that does not belong to BP Q PQ
0 0 1
0 1 1
1 0 0
1 1 1
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@2004Adriano Cruz NCE e IM - UFRJ Geometry of Fuzzy Sets 21
Zadeh´s definition of SubsetsZadeh´s definition of Subsets A is a subset of B iff there is no element
of A that does not belong to B A B iff A(x) B(x) for all x
P Q PQ
0 0 1
0 1 1
1 0 0
1 1 1
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@2004Adriano Cruz NCE e IM - UFRJ Geometry of Fuzzy Sets 22
Subsethood examplesSubsethood examples Consider A={(x1,1/3),(x2=1/2)} and
B={(x1,1/2),(x2=3/4)} A B, but B A
(0,1)
1/2
3/4
(1,1)
(1,0)
x1
x2
1/3 1/2
A
B
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@2004Adriano Cruz NCE e IM - UFRJ Geometry of Fuzzy Sets 23
Not Fuzzy SubsethoodNot Fuzzy Subsethood The so called membership dominated
definition is not fuzzy. The fuzzy power set of B (F(2B)) is the
hyper rectangle docked at the origin of the hyper cube.
Any set is either a subset or not.
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@2004Adriano Cruz NCE e IM - UFRJ Geometry of Fuzzy Sets 24
Fuzzy power set sizeFuzzy power set size F(2B) has infinity cardinality. For finite dimensional sets the size of
F(2B) is the Lebesgue measure or volume V(B)
(0,1)
1/2
3/4
(1,1)
(1,0)
x1
x2
1/3 1/2
A
B
n
iiB xBV
1
)()(
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@2004Adriano Cruz NCE e IM - UFRJ Geometry of Fuzzy Sets 25
Fuzzy SubsethoodFuzzy Subsethood Let S(A,B)=Degree(A B)=F(2B)(A) Suppose only element j violates
A(xj)B(xj), so A is not totally subset of B.
Counting violations and their magnitudes shows the degree of subsethood.
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@2004Adriano Cruz NCE e IM - UFRJ Geometry of Fuzzy Sets 26
Fuzzy SubsethoodFuzzy Subsethood Supersethood(A,B)=1-S(A,B) Sum all violations=max(0,A(xj)-B(xj)) 0S(A,B)1
A
xxBAS
A
xxBAodSupersetho
XxBA
XxBA
))()(,0max(1),(
))()(,0max(),(
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@2004Adriano Cruz NCE e IM - UFRJ Geometry of Fuzzy Sets 27
Subsethood measuresSubsethood measures Consider A={(x1,0.5),(x2=0.5)} and
B={(x1,0.25),(x2=0.9)}
6.0),(5.05.0
))5.09.0(,0max())5.025.0(,0max(1),(
75.0),(
5.05.09.05.0,0max25.05.0,0max
1),(
ABS
ABS
BAS
BAS