1. Geometry of Fuzzy Sets

2. Sets as points Geometry of fuzzy sets includes Domain X={x1,…,x2}
Range of mappings [0,1]
A:X[0,1]
@2004Adriano Cruz
NCE e IM - UFRJ

3. Classic 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}
@2004Adriano Cruz
NCE e IM - UFRJ

4. Vertices 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
@2004Adriano Cruz
NCE e IM - UFRJ

5. The vertices in space
@2004Adriano Cruz
NCE e IM - UFRJ

6. Fuzzy 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
@2004Adriano Cruz
NCE e IM - UFRJ

7. The Fuzzy Cube
@2004Adriano Cruz
NCE e IM - UFRJ

8. Fuzzy 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)}
AA´={(x1,2/3),(x2,3/4)}
AA´={(x1,1/3),(x2,1/4)}
@2004Adriano Cruz
NCE e IM - UFRJ

9. Fuzzy Operations in the Space
@2004Adriano Cruz
NCE e IM - UFRJ

10. Paradox 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 Snot-S and not-SS
t(S)=t(not-S)=1-t(S)  t(S)=0.5
@2004Adriano Cruz
NCE e IM - UFRJ

11. Cardinality 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|
@2004Adriano Cruz
NCE e IM - UFRJ

12. Distance
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
@2004Adriano Cruz
NCE e IM - UFRJ

13. Distance 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
@2004Adriano Cruz
NCE e IM - UFRJ

14. Fuzzy Cardinality
@2004Adriano Cruz
NCE e IM - UFRJ

15. Fuzzy 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.
@2004Adriano Cruz
NCE e IM - UFRJ

16. Fuzzy Entropy Geometry
@2004Adriano Cruz
NCE e IM - UFRJ

17. Fuzzy Operations in the Space
@2004Adriano Cruz
NCE e IM - UFRJ

18. Fuzzy 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…
@2004Adriano Cruz
NCE e IM - UFRJ

19. Subsets Sets contain subsets.
A is a subset of B (AB) 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 (AB iff A2B).
@2004Adriano Cruz
NCE e IM - UFRJ

20. Subsets 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 B P Q
PQ 1
@2004Adriano Cruz
NCE e IM - UFRJ

21. Zadeh´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
PQ 1
@2004Adriano Cruz
NCE e IM - UFRJ

22. Subsethood examples
Consider A={(x1,1/3),(x2=1/2)} and B={(x1,1/2),(x2=3/4)}
A  B, but B  A
@2004Adriano Cruz
NCE e IM - UFRJ

23. Not 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.
@2004Adriano Cruz
NCE e IM - UFRJ

24. Fuzzy 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)
@2004Adriano Cruz
NCE e IM - UFRJ

25. Fuzzy 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.
@2004Adriano Cruz
NCE e IM - UFRJ

26. Fuzzy Subsethood Supersethood(A,B)=1-S(A,B)
Sum all violations=max(0,A(xj)-B(xj))
0S(A,B)1
@2004Adriano Cruz
NCE e IM - UFRJ

27. Subsethood measures
Consider A={(x1,0.5),(x2=0.5)} and B={(x1,0.25),(x2=0.9)}
@2004Adriano Cruz
NCE e IM - UFRJ