A Study of some Important Theorem on Fuzzy Linear Space

Let S be a nonempty set. Then a characteristic function  :S[0,1] is called a fuzzy set of S Let X be a field and be a fuzzy field in X with characteristic function F. Let Y be a linear space over the fuzzy field F and V is a fuzzy subset of Y with characteristic function v. Then V is called a fuzzy linear space in Y if and only if, it satisfied the following condition. (i) v(x+y)min{v( x), v(y)} for all x,yY (ii) v(x) min{v( ), v(x)} for all F and xY (iii) v(o)=1

Let us suppose that Y and Z be vector space (linear space) over a field F in a field X and f be liner transformation from Y in to Z and W be a fuzzy linear space in Z, then the inverse image of W that is f-1(w) be a fuzzy linear space in Y.

Let us assume that Y and Z be two vector space over a fuzzy field F in a field X. Again we assume that f be a Then we take from the Properties of linear transformation that f(x+y) = f(x)+(y) and f(λx) = λf(x)

Again assume that w, be a fuzzy linear space in Z and f-

1[w] be the inverse image of W in Y Then we prove that f-1(w) be a fuzzy linear space in Y for al Then 1()fW(λx+y) = W[f{λ(x) +y}]

= W{λf(x) +f(y)} ≥ min[min{F(λ), Wf(x)},min{F(),Wf(y)}]

=min[min{F1()fW(x)}min{F(

1()fW(y)}] Hence 1()fW(λx+F1()fW(x)}]min{F( µ),1()fW (y)}] Thus it is clear that 1()fWthe inverse image of W in a fuzzy linear spaces in Y.

If V be a fuzzy linear space in a linear spaces Y over an

ordinary filed F in X, then VV

PROOF:-

Let us suppose that V be a fuzzy linear space in a linear space Y over an ordinary field F in X.

VV

Then we have

V(X) = V(EX)= V{(1)X} = V{(1(V

i.e. VV ………………..(II) Thus From (I) And (Ii), We Get

VV

(1) Zadeh, L.A., Fuzzy Sets, Inform and Control 8

(1965) 338-353.

(2) Nanda, S., Fuzzy Fields and Fuzzy Linear Spaces, Fuzzy Sets and System 19 (1986) 89-84. (3) Biswas, R., Fuzzy Fields and Fuzzy Linear Spaces Redefined Fuzzy Sets and System 33 (1989) 257-

259.