EXTENSION OF n - DIMENSIONAL EUCLIDEAN VECTOR SPACE En OVER R TO PSEUDO - FUZZY VECTOR SPACE OVER F 1 p ( 1 )

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 → PQ = (q (1) −p (1) ,q (2) −p (2) ,...,q (n) −p (n)) = Q(−)P. Then we obtain an n-dimensional vector space E n = { → PQ | 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., fuzzy vector space is discussed theoretically. In Katsaras and Liu [2], E denotes a vector space over K, where K is the space of real or complex numbers. A fuzzy set F in E is called a fuzzy subspace if (a) F + F ⊂ F ; (b) λF ⊂ F for every scalar λ. Katsaras and Liu introduced the concept of a fuzzy subspace of a vector space. In Das [1], E denotes a vector space over a field K. Let I = [0, 1] and let I E be the collection of all mappings