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Geometry of Fuzzy Sets

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 |

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Geometry of Fuzzy Sets

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  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] 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} 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 NCE e IM - UFRJ

  5. The vertices in space 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 NCE e IM - UFRJ

  7. The Fuzzy Cube 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)} NCE e IM - UFRJ

  9. Fuzzy Operations in the Space 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 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| 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 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 NCE e IM - UFRJ

  14. Fuzzy Cardinality 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. NCE e IM - UFRJ

  16. Fuzzy Entropy Geometry NCE e IM - UFRJ

  17. Fuzzy Operations in the Space 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… 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). 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 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 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 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. 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) 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. 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 NCE e IM - UFRJ

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

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