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