1 / 27

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 |

stephanie-gutierrez
Download Presentation

Geometry of Fuzzy Sets

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related