Fuzzy bases and the fuzzy dimension of fuzzy vector spaces

In this paper, new definitions of a fuzzy basis and a fuzzy dimension for a fuzzy vector space are presented. A fuzzy basis for a fuzzy vector space (E, mu) is a fuzzy set beta on E. The cardinality of a fuzzy basis beta is called the fuzzy dimension of (E, mu). The fuzzy dimension of a finite dimensional fuzzy vector space is a fuzzy natural number. For a fuzzy vector space, any two fuzzy bases have the same cardinality. If (E) over tilde (1) and (E) over tilde (2) are two fuzzy vector spaces, then dim ((E) over tilde (1) + (E) over tilde (2)) + dim ((E) over tilde (1) boolean AND (E) over tilde (2)) = dim((E) over tilde (1)) + dim((E) over tilde (2)) and dim((ker) over tildef) + dim((im) over tildef) = dim((E) over tilde) hold without any restricted conditions