Extension of n-dimensional Euclidean vector space En over ℝ to pseudo-fuzzy vector space over Fp1(1)

Abstract

For any two points P=(p(1),p(2),…,p(n)) and Q=(q(1),q(2),…,q(n)) of ℝn, we define the crisp vector PQ⟶=(q(1)−p(1),q(2)−p(2),…,q(n)−p(n))=Q(−)P. Then we obtain an n-dimensional vector space En={PQ⟶|   for all   P,Q∈ℝn}. Further, we extend the crisp vector into the fuzzy vector on fuzzy sets of ℝn. Let D˜,E˜ be any two fuzzy sets on ℝn and define the fuzzy vector E˜D˜⟶=D˜⊖E˜, then we have a pseudo-fuzzy vector space.