Geometry of Fuzzy Sets

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

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

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

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

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

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

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

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

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

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

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

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

Fuzzy Cardinality @2004Adriano Cruz NCE e IM - UFRJ

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

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

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

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

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

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

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

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

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

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

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

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

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