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

PII. S0161171203007932 http://ijmms.hindawi.com © Hindawi Publishing Corp.

EXTENSION OF

n

-DIMENSIONAL EUCLIDEAN VECTOR

SPACE

E

OVER

R

TO PSEUDO-FUZZY VECTOR

SPACE OVER

F

p

(

1

)

KWEIMEI WU

Received 17 July 2001

For any two pointsP=(p(1)_,p(2)_,...,p(n)_)andQ=(q(1)_,q(2)_,...,q(n)_)ofRn_{, we} define the crisp vectorP Q→=(q(1)−p(1)_,q(2)_−p(2)_,...,q(n)_−p(n)_{)=Q(−)P. Then} we obtain ann-dimensional vector spaceEn_{= {P Q}→_{| for allP,Q∈R}n_{}. Further,} we extend the crisp vector into the fuzzy vector on fuzzy sets ofRn_{. LetD,Ebe} any two fuzzy sets onRn_{and define the fuzzy vectorE}→_{D=D E, then we have a} pseudo-fuzzy vector space.

2000 Mathematics Subject Classification: 08A72.

1. Introduction. In [1,2,4,5,6], fuzzy vector space is discussed theoreti-cally. In Katsaras and Liu [2],Edenotes a vector space overK, whereKis the space of real or complex numbers. A fuzzy setFinEis called a fuzzy subspace if (a)F+F ⊂F; (b)λF⊂F for every scalarλ. Katsaras and Liu introduced the concept of a fuzzy subspace of a vector space. In Das [1],Edenotes a vector space over a fieldK. LetI=[0,1]and letIE _{be the collection of all mappings}

ofE intoI. We sayµ∈IE _{is a fuzzy space ofE under a triangular normT}

(see [1, Definition 2.1]), or simply aT-fuzzy subspace ofE if for allx,y∈E

and for alla∈K, µ(x+y)≥T (µ(x),µ(y))andµ(ax)≥µ(x), respectively. In Lubczonok [5], a fuzzy vector space is a pairE=(E,µ), whereEis a vector space andµ:E→[0,1]with the property that, for alla,b∈Randx,y∈E, we haveµ(ax+by)≥µ(x)∧µ(y). In Kumar [4],V is a vector space overF, whereF is the field of real numbers. A fuzzy subsetµ ofV is called a fuzzy subspace if it has the following properties:

(a) µ(v1−v2)≥min(µ(v1),µ(v2))for allv1,v2∈V;

(b) µ(αv)≥µ(v)for allα∈F,v∈V.

There are various definitions of fuzzy vector spaces in these papers. All of them use the fuzzy setµover a crisp vector spaceE, orµ:E→[0,1], to define fuzzy vector. These are different from our work. Pick two pointsP=(p1,p2,...,pn),

Q=(q1,q2,...,qn)inRnto form a vectorP Q→=(q1−p1,q2−p2,...,qn−pn).

Then extend this vector to the fuzzy vectorP→Q=QPformed by fuzzy setsP,

D(α)

x(1)

0

α

1.0

z

z=µD˜

x(1),x(2)

x(2)

Figure_2.1. _{α-cut of fuzzy set}DonR2_.

Section 2 is a preparing work. Section 3 is the extension of the crisp n -dimensional Euclidean vectorEn _{to the pseudo-fuzzy vector space. We talk}

inSection 4 about the length of the fuzzy vectors and fuzzy inner product.

Section 5is a more discussion.

2. Preparation. In order to consider the fuzzy vectors of fuzzy sets onRn_,

we ought to know the following. First, from Kaufmann and Gupta [3] and Zim-mermann [8], we have the following definition.

Definition2.1. (a) A fuzzy setAonR=(−∞,∞)is convex if and only if

every ordinary setA(α)= {x|µA(x)≥α}for allα∈[0,1]is convex. Thus

A(α)is a closed interval inR.

(b) A fuzzy setAonRis normal if and only ifx∈RµA(x)=1.

We can extend this definition to Rn_{, say if D is a fuzzy set on R}n _with

membership function

µD

x(1),x(2),...,x(n)∈[0,1] ∀x(1),x(2),...,x(n)∈Rn_{, (2.1)}

then we have the following definition.

Definition2.2. Theα-cut of fuzzy setDonRn, 0≤α≤1, is defined by

D(α)=x(1),x(2),...,x(n)|µD

x(1),x(2),...,x(n)≥α. (2.2)

EXTENSION OFn-DIMENSIONAL EUCLIDEAN VECTOR SPACEE ... 2351

Definition_2.3. _{(a) A fuzzy setD on}Rn_{is convex if and only if for each}

α∈[0,1], every ordinary set

D(α)=x(1)_,x(2)_,...,x(n)_|µ

D

x(1)_,x(2)_,...,x(n)_{≥α (2.3)}

is a convex closed subset ofRn_.

(b) A fuzzy setDis normal if and only if

(x(1)_,x(2)_,...,x(n)_)∈Rn

µD

x(1)_,x(2)_,...,x(n)_{=1. (2.4)}

LetFcbe the family of all fuzzy sets onRn_{satisfyingDefinition 2.3(a), (b).}

Remark2.4. Whenα=0, then theα-cut is

x(1)_,x(2)_,...,x(n)_|µ

D

x(1)_,x(2)_,...,x(n)_{≥0. (2.5)}

LetD(0)be the smallest convex closed subset inRn_satisfying

x(1)_,x(2)_,...,x(n)_|µ

D

x(1)_,x(2)_,...,x(n)_{≥0 (2.6)}

(seeExample 4.11).

Definition_{2.5(Pu and Liu [7])}. _{If the membership function of a fuzzy set}

aα, 0≤α≤1, onRis

µaα=

 

α, x_{0, x}=₌a,_a, (2.7)

then we callaαa level-αfuzzy point onR.

LetF_p(α)1 _{= {aα| for alla∈R}be the family of all level-αfuzzy points on} Rsatisfying (2.7).

Definition_2.6. _{If the membership function of a fuzzy set(a}(1)_,a(2)_,...,

a(n)_{)α, 0≤α≤1, onR}n_is

µ_(a(1_),a(2_{),...,a(n))α}

x(1),x(2),...,x(n)

=

  α,

ifx(1)_,x(2)_,...,x(n)_=a(1)_,a(2)_,...,a(n)_,

0, elsewhere,

(2.8)

then we call(a(1)_,a(2)_,...,a(n)_{)αa level-αfuzzy point onR}n_.

LetFn

p(α)= {(a(1),a(2),...,a(n))α| for all(a(1),a(2),...,a(n))∈Rn}be the

For everyaα ∈F_p(α)1 _{, let aα=(a,a,...,a)α, then aα can be regarded as}

a special case of the level-αfuzzy point(a(1)_,a(2)_,...,a(n)_{)αdegenerating to} a(1)_=a(2)_{= ··· =a}(n)_{=a. Thus}

µ(a,a,...,a)α

x(1),x(2),...,x(n)=     

α, x(1)_,x(2)_,...,x(n)_{=(a,a,...,a),}

0, x(1)_,x(2)_,...,x(n)_=(a,a,...,a)

=µaα

x(1),x(2),...,x(n).

(2.9)

Remark 2.7. We can regard aα as a fuzzy set onRas the form in (2.7)

or it can also be regarded as a fuzzy set onRn _{such asaα=(a,a,...,a)α in}

(2.9) according to how we want it to be. That is, 01=(0,0,...,0)1 and aα=

(a,a,...,a)α,α∈[0,1].

From Kaufmann and Gupta [3], forD,E⊂Rn_{,k∈R, we have}

(i) D(+)E={(x(1)_+y(1)_,x(2)_+y(2)_,...,x(n)_+y(n)_{)| ∀(x}(1)_,x(2)_,...,x(n)_)∈ D, (y(1)_,y(2)_,...,y(n)_)∈E},

(ii) D(−)E={(x(1)_−y(1)_,x(2)_−y(2)_,...,x(n)_−y(n)_{)| ∀(x}(1)_,x(2)_,...,x(n)_)∈ D, (y(1)_,y(2)_,...,y(n)_)∈E},

(iii) k(·)D= {(kx(1)_,kx(2)_,...,kx(n)_{)| ∀(x}(1)_,x(2)_,...,x(n)_)∈D},

(iv) theα-cut ofD⊕EisD(α)(+)E(α), (v) theα-cut ofDEisD(α)(−)E(α), (vi) theα-cut ofk1D isk(·)D(α).

3. The extension of the crispn-dimensional Euclidean vector spaceEn_to the pseudo-fuzzy vector spaceSFR. In crisp case, forP=(p(1)_,p(2)_,...,p(n)_), Q=(q(1)_,q(2)_,...,q(n)_),A=(a(1)_,a(2)_,...,a(n)_{), B=(b}(1)_,b(2)_,...,b(n)_)∈Rn_,

andk∈R, we can define the operations “+,·” for the crisp vectorsP Q→,AB→in

En_{, then-dimensional vector space overR}n_{, by}

→

AB=b(1)_−a(1)_,b(2)_−a(2)_,...,b(n)_−a(n)_,

→

P Q=q(1)−p(1),q(2)−p(2),...,q(n)−p(n),

(3.1)

→

AB+P Q→=b(1)+q(1)−a(1)−p(1),b(2)+q(2)

−a(2)_−p(2)_,...,b(n)_+q(n)_−a(n)_−p(n)_,

k·P Q→=kq(1)_−kp(1)_,kq(2)_−kp(2)_,...,kq(n)_−kp(n)_.

(3.2)

LetO=(0,0,...,0)∈Rn_{, thenOP}→_=(p(1)_,p(2)_,...,p(n)_)andOO→_{=(0,0,...,0)∈} En_.

LetEn_{be ann-dimensional vector space overR. ByDefinition 2.6,F}n p(1)=

{(a(1)_,a(2)_,...,a(n)_{)1| for all(a}(1)_,a(2)_,...,a(n)_)∈Rn_{}. This is a family of all}

EXTENSION OFn-DIMENSIONAL EUCLIDEAN VECTOR SPACEE ... 2353

We notice that there is a one-to-one onto mappingρbetween(a(1)_,a(2)_,..., a(n)_)∈Rn_and(a(1)_,a(2)_,...,a(n)₎

1∈Fpn(1). That is,

ρ:a(1)_,a(2)_,...,a(n)_∈Rn_←→ρa(1)_,a(2)_,...,a(n)

=a(1),a(2),...,a(n)1∈Fpn(1),

(3.3)

µ_(a(1_),a(2_),...,a(n))1

x(1)_,x(2)_,...,x(n)

=C_(a(1_),a(2_),...,a(n))

x(1),x(2),...,x(n),

(3.4)

whereCAis the characteristic function ofA.

Let P=(p(1)_,p(2)_,...,p(n)_{)1, Q =(q}(1)_,q(2)_,...,q(n)_)1∈Fn

p(1). From (3.1),

(3.3), we have the following definition:

→

PQ=q(1)_−p(1)_,q(2)_−p(2)_,...,q(n)_−p(n)

1=QP . (3.5)

We callP→Qa fuzzy vector.

LetO=(0,0,...,0)1∈Fpn(1), then

→

OP=(p(1)_,p(2)_,...,p(n)₎

1, →

OO=(0,0,...,

0)1. LetFEn= { →

PQ| for allP ,Q∈Fn

p(1)}be the family of all fuzzy vectors on Fn

p(1). From (3.1), (3.5), we can have the one-to-one onto mappingρbetween En_andFEn_by

→

P Q=q(1)_−p(1)_,q(2)_−p(2)_,...,q(n)_−p(n) _∈En_{←→ρP Q}→

=q(1)_−p(1)_,q(2)_−p(2)_,...,q(n)_−p(n)

1

=P→Q∈FEn_.

(3.6)

Since(p(1)_,p(2)_,...,p(n)_)=OP→_{, hence the point inR}n_{can be regarded as a}

vec-tor inEn_{. Also since(p}(1)_,p(2)_,...,p(n)₎

1= →

OP, hence the level-1 fuzzy points onRn_{can be regarded as the fuzzy vectors inFE}n_{. Therefore the mapping in}

(3.3) is a special case of the mapping in (3.6).

The operations “⊕,” of the fuzzy vectors inFEn_{have the following}

prop-erty.

Property _3.1. _ForP=_(p(1)_,p(2)_,...,p(n)_{)1, Q}=_(q(1)_,q(2)_,...,q(n)_)1,A=

(a(1)_,a(2)_,...,a(n)₎

1,B=(b(1),b(2),...,b(n))1∈FEn, andk=0∈R, we have

→

AB⊕P→Q=b(1)_+q(1)_−a(1)_−p(1)_,b(2)_+q(2)_−a(2)_−p(2)_,...,b(n)

+q(n)−a(n)−p(n)1;

(3.7)

k1 →

Proof. _{For (3.7),}

µ→

AB⊕P→Q

z(1)_,z(2)_,...,z(n)

= sup

z(j)_=x(j)_+y(j)_,j=1,2,...,n

µ_(b(1)−a(1_),b(2)−a(2_{),...,b(n)−a(n))1}

x(1)_,x(2)_,...,x(n)

∧µ_(q(1)−p(1_),q(2)−p(2_{),...,q(n)−p(n))1}

y(1),y(2),...,y(n)

= sup

(y(1)_,y(2)_,...,y(n)₎

µ_(b(1)−a(1_),b(2)−a(2_{),...,b(n)−a(n))1}

z(1)−y(1),

z(2)_−y(2)_,...,z(n)_−y(n)

∧µ_(q(1)−p(1_),q(2)−p(2_{),...,q(n)−p(n))1}

y(1),y(2),...,y(n)

=1, ifz(j)_−y(j)_=b(j)_−a(j)_{, y}(j)_=q(j)_−p(j)_{, j=1,2,...,n,}

=1, ifz(j)_−q(j)_+p(j)_=b(j)_−a(j)_{, j=1,2,...,n,}

=µ_(b(1)+q(1)−a(1)−p(1_),b(2)+q(2)−a(2)−p(2_{),...,b(n)+q(n)−a(n)−p(n))}

z(1),z(2),...,z(n)

∀z(1)_,z(2)_,...,z(n)_∈Rn_,

(3.9)

that is,

→

AB⊕P→Q=b(1)_+q(1)_−a(1)_−p(1)_,b(2)_+q(2)_−a(2)

−p(2)_,...,b(n)_+q(n)_−a(n)_−p(n)

1.

(3.10)

Similarly, we have (3.8). In the case k =0, it follows by Property 3.7(7). FromProperty 3.1, (3.2), (3.6), (3.7), and (3.8), we have

ρAB→+P Q→=b(1)_+q(1)_−a(1)_−p(1)_,b(2)_+q(2)

−a(2)_−p(2)_,...,b(n)_+q(n)_−a(n)_−p(n)

1

=A→B⊕P→Q=ρAB→⊕ρP Q→,

ρk·AB→=k1 →

AB=ρ(k)ρ _→

AB

.

(3.11)

ByRemark 2.7,k=(k,k,...,k). Hence by (3.3),ρ(k)=ρ(k,k,...,k)=(k,k,..., k)1=k1. From (3.6), (3.11), sinceEn is a vector space overR, therefore FEn

satisfies the conditions to be a vector space too. We callFEn_{a fuzzy vector}

space overF_p(1 _{1)and callP}→_{Q ( ∈FE}n_{)a fuzzy vector.}

Remark_3.2. _{The zero fuzzy vector} →

OO=(0,0,...,0)1inFEn will be

ob-tained from the zero vectorOO→=(0,0,...,0)inEn_{by mappingOO}→_toO→_O.

Obviously, there is a one-to-one mapping betweenRand F1

EXTENSION OFn-DIMENSIONAL EUCLIDEAN VECTOR SPACEE ... 2355

Property_3.3. _{The fuzzy vector spaceFE}n_overFp(1 _{1)is equivalent to the}

vector spaceEn_{overRdenoted byE}n_≈FEn_.

Since theα-cut of the fuzzy pointP=(p(1)_,p(2)_,...,p(n)₎

1inFpn(1)is(p(1), p(2)_,...,p(n)_{)for allα∈[0,1], hence it can be regarded as a special case inFc,}

that is, we can takeFn

p(1)as a subfamily ofFc, that is,Fpn(1)⊂Fc. Therefore we

can extend the fuzzy vector spaceFEn_{toFcand have the following definition}

similarly as in (3.5).

Definition3.4. ForX, Y∈Fc, define →

XY=YX. (3.12)

We callX→Ya fuzzy vector.

LetSFR= {X→Y=YX| for allX, Y∈Fc}.

Property_3.5. _For →

XY , W→Z∈SFR,

→

XY=W→Z iffYX=ZW . (3.13)

Proof. _{The proof follows fromDefinition 3.4of fuzzy vector.}

Property3.6. For →

XY , W→Z∈SFR,k∈R, (1) X→Y⊕W→Z=A→B; hereA=X⊕W,B=Y⊕Z;

(2) k1 →

XY=C→D; hereC=k1X,D=k1Y.

Proof. (1) For eachα∈[0,1], from (i), (ii), (iv), (v), theα-cuts of →

XY=YX,

→

WZ=ZWareY (α)(−)X(α),Z(α)(−)W (α), respectively. Let

D=x(1)_,x(2)_,...,x(n)_{∈X(α), y}(1)_,y(2)_,...,y(n)_{∈Y (α),}

z(1),z(2),...,z(n)∈Z(α),w(1),w(2),...,w(n)∈W (α). (3.14)

Therefore theα-cut ofX→Y⊕W→Zis

Y (α)(−)X(α)(+)Z(α)(−)W (α)

=y(1)_−x(1)_+z(1)_−w(1)_,y(2)_−x(2)_+z(2)_−w(2)_,..., y(n)_−x(n)_+z(n)_−w(n)_|D

=y(1)+z(1),y(2)+z(2),...,y(n)+z(n)

|y(1),y(2),...,y(n)∈Y (α),z(1),z(2),...,z(n)∈Z(α) (−)x(1)+w(1),x(2)+w(2),...,x(n)+w(n)

|x(1),x(2),...,x(n)∈X(α), w(1),w(2),...,w(n)∈W (α)

(3.15)

Property_3.7. _For →

XY , W→Z,U→V∈SFR,k,t∈R, (1) X→Y⊕W→Z=W→Z⊕X→Y;

(2) (X→Y⊕W→Z) ⊕U→V=X→Y⊕(W→Z⊕U→V ) ;

(3) X→Y⊕O→O=X→Y;

(4) k1(t1 →

XY ) =(kt)1 →

XY; (5) k1(

→

XY⊕W→Z) =(k1 →

XY ) ⊕(k1 →

WX) ; (6) 11

→

XY=X→Y; (7) 01

→

XY=O→O.

Proof. _{For eachα}∈_{[0,1]and from (iv), (v), and (vi),}

(1) theα-cut ofX→Y⊕W→Z=(YX) ⊕(Z−W ) is

Y (α)(−)X(α)(+)Z(α)(−)W (α)

=Z(α)(−)W (α)(+)Y (α)(−)X(α) (3.16)

which is theα-cut ofW→Z⊕X→Y. Therefore (1) holds; (2) the proof is similar to (1);

(3) sinceO→O=(0,0,...,0)1, theα-cut of →

XY⊕O→O=(YX) ⊕(0,0,...,0)1

is(Y (α)(−)X(α))(+)(0,0,...,0)=Y (α)(−)X(α)which is theα-cut of

→

XY. ThereforeX→Y⊕O→O=X→Y;

(4) for eachα∈[0,1], theα-cut ofk1(t1 →

XY ) =k1(t1(YX)) is

k(·)(t(·)(Y (α)(−)X(α)))=(kt)(·)(Y (α)(−)X(α)) which is theα-cut of(kt)1(·)

→

XY;

(5) the proof is similar to (4); (6) the proof is similar to (5);

(7) from (v), (vi), for eachα∈[0,1], theα-cut of 01 →

XYis(0,0,...,0)which is theα-cut ofO→O.

In order to be a fuzzy vector space, it needs that the following hold:

(8) for anyX→Y∈SFR, there existsW→Z∈SFRsuch thatX→Y⊕W→Z=O→O,

(9) (m+n)1 →

XY=(m1 →

XY ) ⊕(n1 →

XY ) for allX→Y∈SFR.

Now sinceFEn_{is a vector space, (8), (9) hold without any question. IfX, Y∈}

Fc, but∈Fn p(1)and

→

XY=O→O,W→Z=O→O. By (iv), (v), for eachα∈[0,1]the

α-cuts of X→Y and W→Zare Y (α)(−)X(α)=(0,0,...,0) and Z(α)(−)W (α)= (0,0,...,0), respectively. Theα-cut ofX→Y⊕W→Z=(YX) ⊕(ZW ) is

Y (α)−X(α)(+)Z(α)(−)W (α)=(0,0,...,0), (3.17)

EXTENSION OFn-DIMENSIONAL EUCLIDEAN VECTOR SPACEE ... 2357

ForX, Y∈Fc but∈Fn

p(1), from (i), (ii), (iii), (iv), (v), and (vi), theα-cut(0≤ α≤1)of(m+n)1

→

XY=(m+n)1(YX) is

(m+n)Y (α)(−)(m+n)X(α)

=(m+n)y(1)−(m+n)x(1),(m+n)y(2)−(m+n)x(2),..., (m+n)y(n)−(m+n)x(n)|x(1),x(2),...,x(n)∈X(α),

y(1)_,y(2)_,...,y(n)_{∈Y (α).}

(3.18)

However, theα-cut of(m1 →

XY ) ⊕(n1 →

XY ) =(m1(YX)) ⊕(n1(YX))

is

mY (α)(−)X(α)(+)nY (α)(−)X(α)

=my(1)_−x(1)_+ny(1)_−x(1)_,my(2)_−x(2)

+ny(2)_−x(2)_,...,my(n)_−x(n)_+ny(n)_−x(n)

|x(1),x(2),...,x(n),x(1),x(2),...,x(n)∈X(α),

y(1),y(2),...,y(n),y(1),y(2),...,y(n)∈Y (α).

(3.19)

Therefore(m+n)y(j)_−(m+n)x(j) _=m(y(j)_−x(j)_)+n(y(j)_−x(j)_{) if} (x(j)_=x(j) _orx(j)_)or(y(j)_=y(j)_ory(j)_{). Hence (9) does not hold.}

Definition_3.8. _{TheSFRwhich satisfiesProperty 3.7(1), (2), (3), (4), (5), (6),}

and (7) is called pseudo-fuzzy vector space overF_p(1 _{1), and callX}→_{Y ( ∈SFR)a}

fuzzy vector.

ThenEn_≈FEn_{⊂SFR. That is, we can regardSFRas an extension ofE}n_,

but only obtain a pseudo-fuzzy vector space instead of a fuzzy vector space. Its addition⊕and multiplicationare followed byProperty 3.6.

Property_3.9. _For

→

XjYj ∈SFR,a(j)1 ∈Fp(1 1),j=1,2,...,m,

a(11)

→

X1Y1

⊕a(12)

→

X2Y2

⊕···⊕a(m)1

→

XmYm =A→B; (3.20)

hereA=C1⊕C2⊕ ··· ⊕Cm ,B=D1⊕D2⊕ ··· ⊕Dm , andCj =a(j)1 Xj,Dj =

a(j)1 Yj,j=1,2,...,m.

Proof. _{The proof follows fromProperty 3.6(1), (2) and mathematical}

in-duction.

Property3.10. ForY∈Fc, butY∉Fn

p(1),Y=O, andX∈Fc,

(1) YY=O;

(3) X→Y=YX=O→O;

(4) Y→X=XY=O→O.

Proof. (1) SinceY =O, theα-cut of Y is Y (α)=(0,0,...,0)for allα∈

[0,1]. By (ii), (v), theα-cut ofYYis

Y (α)(−)Y (α)

=s(1)−t(1),s(2)−t(2),...,s(n)−t(n)|s(1),s(2),...,s(n),

t(1),t(2),...,t(n)∈Y (α)=(0,0,...,0).

(3.21)

ThereforeYY=O.

(2) The proof follows by (1). (3) The proof is similar to (1). (4) The proof is similar to (1).

Remark_3.11. _{(a) IfY}=_(p(1)_,p(2)_,...,p(n)₎₁∈_Fn

p(1), thenYY=(p(1)−

p(1)_,p(2)_−p(2)_,...,p(n)_−p(n)_{)1=(0,0,...,0)1. HenceY}→_Y=O→_O.

(b) It is trivial thatO→O⊕O→O=O→O.

(c) LetSFV= {P→X| for allP=(p(1)_,p(2)_,...,p(n)₎

1,∈Fpn(1),X∈Fc}, then En_≈FEn_{⊂SFV⊂SFR.}

Property3.12. ForX∈Fc,

(1) 01X=O;

(2) O⊕X=X.

Proof. _{The proof is obvious.}

Example_3.13(n=₂₎. _{A car carrying a rocket departs from pointQ}=_(1,2)

passes through pointS=(5,8), arrives at pointW=(10,15), and launches the rocket from there. Suppose its target is located atZ=(100,200). Chances are the rocket will not hit atZexactly. Instead it would probably drop in the vicinity ofZ. Let

O(100,200),1=(x,y)|(x−100)2+(y−200)2≤1, (3.22)

and the point hit isZ,(Z∈Fc)with membership function

µZ(x,y)=

  

1−(x−100)2_−(y−200)2_{, if(x,y)∈O(100,200),1,}

0, if(x,y)∈O(100,200),1. (3.23)

Theα-cut(0≤α≤1)ofZis

Z(α)=(x,y)|µZ(x,y)≥α

EXTENSION OFn-DIMENSIONAL EUCLIDEAN VECTOR SPACEE ... 2359

V Z U

O((100,200),1)

Q(1,2)

Figure_3.1. _{Fuzzy vector}Q→Z.

If we choose the route from base to target to beQ→S→W→Z, then we have the crisp vectorsQS→=(4,6),SW→=(5,7), andW Z→=(90,185). So the crisp vector fromQtoZ isQS→+SW→+W Z→=(99,198). And the route in the fuzzy sense is

Q=(1,2)1 →S=(5,8)1 →W=(10,15)1 →Z (3.25)

with (3.23) as the membership function ofZ. We then have fuzzy vectorsQ→S= (4,6)1,

→

SW=(5,7)1, and →

WZ=ZW. From (3.23), the membership function ofW→Zis

µ→

WZ(x,y)=µZ(x+10,y+15)

=

  

1−(x−90)2_−(y−185)2_{, if(x−90)}2_+(y−185)2_≤1,

0, elsewhere.

(3.26)

The fuzzy vector from base_→ Qto targetZbyProperty 3.6(1) isQ→S⊕SW→⊕W→Z=

QZ, with membership function

µ→

QZ(x,y)=µZ(x+1,y+2)

=

  

1−(x−99)2_−(y−198)2_{, if(x−99)}2_+(y−198)2_≤1,

0, elsewhere.

(3.27)

LetZ=(100,200),U=(99.5,200.5), andV=(100.2,199.7)∈O((100,200),1).

As shown inFigure 3.1, the crisp vectors fromQtoZ,U,VinO((100,200),1)

→

QZare

µ→

QZ

_→ QZ=µ→

QZ(99,198)=1,

µ→

QZ

_→ QU=µ→

QZ(98.5,198.5)=0.5,

µ→

QZ

_→ QV=µ→

QZ(99.2,197.7)=0.87.

(3.28)

4. The length of fuzzy vectors inSFRand fuzzy inner product

4.1. The length of fuzzy vectors inSFR. LetP=(p(1)_,p(2)_,...,p(n)_),Q= (q(1)_,q(2)_,...,q(n)_)∈Rn_{. The vectorP Q}→_inEn_{,P Q}→_=(q(1)_−p(1)_,q(2)_−p(2)_,..., q(n)_−p(n)_{)=Q(−)P, has length|P Q}→_{| =}n

j=1(q(j)−p(j))2, called the length

of the vector P Q→. Now let P =(p(1)_,p(2)_,...,p(n)_{)1 and Q = (q}(1)_,q(2)_,...,

q(n)₎

1∈Fpn(1). SinceFEn≈En, we can define the length of fuzzy vector

→

PQ=

(q(1)_−p(1)_,q(2)_−p(2)_,...,q(n)_−p(n)₎

1by| →

PQ| = |P Q→| =n

j=1(q(j)−p(j))2.

SinceFEn_{⊂SFR, we may extend this thought toSFR. Similar to the fuzzy}

vectors inFEn_{, for the vectorX}→_{Y=YX∈SFR, itsα-cut(0≤α≤1)is}

Y (α)(−)X(α)=y(1)−x(1),y(2)−x(2),...,y(n)−x(n)

|x(1)_,x(2)_,...,x(n)_∈X(α),

y(1)_,y(2)_,...,y(n)_{∈Y (α),}

(4.1)

whereX(α),Y (α)are theα-cuts ofX,Y, respectively. For each pointPX(α)=

(x(1)_,x(2)_,...,x(n)_{)∈X(α)andPY(α)=(y}(1)_,y(2)_,...,y(n)_{)∈Y (α), the crisp}

vectorPX(α)PY(α)→=(y(1)_−x(1)_,y(2)_−x(2)_,...,y(n)_−x(n)_{)has length}

PX(α)PY(α)→=

n

j=1

y(j)_−x(j)2 _(4.2)

which is the distance between two pointsPX(α)andPY(α). For any(x(1)_,x(2)_, ...,x(n)_{)∈X(α), denoted simply by(x}(j)_{)∈X(α), let}

d∗Y (α)(−)X(α)= sup

(x(j)_)∈X(α),(y(j)_{)∈Y (α)}

PX(α)PY(α)→

= sup

(x(j))∈X(α),(y(j))∈Y (α) n

j=1

y(j)−x(j)2.

(4.3)

SinceX,Y∈Fc, and byDefinition 2.3(a),X(α),Y (α)are convex closed sub-sets ofRn_{, sod}∗_{(Y (α)(−)X(α))exists. Andd}∗_{(Y (α)(−)X(α))is the longest}

EXTENSION OFn-DIMENSIONAL EUCLIDEAN VECTOR SPACEE ... 2361

Definition_4.1. _For →

XY∈SFR, define the length ofX→Yto be

_X→_Y∗= sup

0≤α≤1d

∗_{Y (α)(−)X(α). (4.4)}

Property_4.2. _For →

XY∈SFR, let the 0-cuts (α-cuts,α=0) ofX,YbeX(0),

Y (0) by Remark 2.4. Then there exist (x(m1)(0),xm(2)(0),...,xm(n)(0))∈ X(0), (ym(1)(0),ym(2)(0),...,ym(n)(0))∈Y (0)such that

_X→_Y∗= sup

0≤α≤1d

∗_{Y (α)(−)X(α)=}

n

j=1

ym(j)(0)−x(j)m(0)2. (4.5)

Proof. Let theα-cuts(0≤α≤1)ofX,Ybe

X(α)=x(1)_,x(2)_,...,x(n)_|µ

X

x(1)_,x(2)_,...,x(n)_≥α,

Y (α)=y(1)_,y(2)_,...,y(n)_|µ

Y

y(1)_,y(2)_,...,y(n)_≥α. (4.6)

It is obvious thatX(α)⊂X(β),Y (α)⊂Y (β)if 0≤β≤α≤1. Since

d∗Y (α)(−)X(α)= sup

(x(j)_)∈X(α),(y(j)_{)∈Y (α)}

n

j=1

y(j)_−x(j)2

, (4.7)

we have

d∗(Y α)(−)X(α)≤d∗Y (β)(−)X(β) ∀0≤β≤α≤1. (4.8)

So

sup

0≤α≤1d

∗_{Y (α)(−)X(α)}

= sup

(x(j)(0))∈X(0),(y(j)(0))∈Y (0) n

j=1

y(j)(0)−x(j)(0)2.

(4.9)

SinceX,Y∈Fc, by the definition ofFc, we know that X(0),Y (0)are convex closed subsets ofRn_{. Hence there exist(x}(1)

m (0),x(m2)(0),...,x(n)m (0))∈X(0), (ym(1)(0),ym(2)(0),...,ym(n)(0))∈Y (0)such that

_X→_Y∗= sup

0≤α≤1d

∗_{Y (α)(−)X(α)=}

n

j=1

ym(j)(0)−xm(j)(0)2. (4.10)

Example4.3. InExample 3.13, the rocket ejected atWtakes the routeQ=

function of fuzzy targetZis

µZ(x,y)=

  

1−(x−100)2_−(y−200)2_{, if(x−100)}2_+(y−200)2_≤1,

0, elsewhere.

(4.11)

We obtain Fuzzy vectorsQ→S=(4,6)1, →

SW=(5,7)1, →

WZ, andQ→Z. The for-mer two have lengths |Q→S|∗ ₌√₄2₊₆2_{=7.21 and|SW}→_|∗ ₌√₅2₊₇2_=8.6,

respectively. As for the length ofW→Z, since for eachα∈[0,1], theα-cuts of

W, ZareW (α)=(10,15),Z(α)= {(x,y)|(x−100)2_+(y−200)2_≤1−α},

respectively. The longest distance between pointsPW (α)=(10,15)∈W (α)and

P_Z(α)=(x,y)∈Z(α)is

d∗Z(α)(−)W (α)=(100−10)2_+(200−15)2_+1−α

=42325+1−α=205.73+1−α. (4.12)

Hence byDefinition 4.1|W→Z|∗_=sup

0≤α≤1d∗(Z(α)(−)W (α))=206.73.

Similarly, we can calculate the length ofQ→Z: for eachα∈[0,1], theα-cut ofQ isQ(α)=(1,2), so

d∗Z(α)(−)Q(α)=(100−1)2_+(200−2)2_+1−α

=49005+1−α=221.37+1−α. (4.13)

Therefore|Q→Z|∗_=sup

0≤α≤1d∗(Z(α)(−)Q(α))=222.37.

Remark_4.4. _{By Cauchy-Schwartz inequality}

a1b1+a2b2+···+anbn2≤a21+a22+···+a2n

b21+b22+···+b2n , (4.14) we have n

j=1

ajbj≤ n

j=1

ajbj ≤ n

j=1

a2j n

j=1

b2j. (4.15)

Therefore,

n

j=1

a2j+ n

j=1

bj2+2 n

j=1

ajbj≤

n

j=1

a2j+ n

j=1

b2j+2 n

j=1

a2j n

j=1

b2j, (4.16)

that is, n

j=1

aj+bj2≤

n

j=1

a2 j+ n

j=1

b2

EXTENSION OFn-DIMENSIONAL EUCLIDEAN VECTOR SPACEE ... 2363

Property_4.5. _For →

XY , U→V∈SFR,k1∈Fp(1 1),k=0,

(1) |k1 →

XY|∗_{= |k|·|X}→_Y|∗_;

(2) |X→Y⊕W→Z|∗_{≤ |X}→_Y|∗_+|W→_Z|∗_.

Proof. _{(1) For eachα}∈_{[0,1], theα-cut ofk1} →

XY=k1(Y ( −)X) is

k(·)Y (α)(−)X(α)

=ky(1)_−kx(1)_,ky(2)_−kx(2)_,...,ky(n)_−kx(n)

|x(1)_,x(2)_,...,x(n)_{∈X(α), y}(1)_,y(2)_,...,y(n)_{∈Y (α).}

(4.18)

For eachα∈[0,1],

d∗k(·)Y (α)(−)X(α)

= sup

(x(j))∈X(α),(y(j))∈Y (α) n

j=1

ky(j)−kx(j)2

= |k|d∗Y (α)−X(α).

(4.19)

Therefore,

k1

→

XY∗= sup

0≤α≤1d

∗_{k(·)Y (α)(−)X(α)}

= |k|sup

0≤α≤1d

∗_{Y (α)(−)X(α)= |k|·X}→_Y∗_.

(4.20)

(2) FromProperty 3.6(1),X→Y⊕W→Z=A→B=BA, whereA=X⊕W,B=Y⊕Z. For eachα∈[0,1], theα-cuts ofX,Y,W, andZareX(α),Y (α),W (α), and

Z(α), respectively, and theα-cut ofBAis

Y (α)(+)Z(α)(−)X(α)(+)W (α)

=y(1)_+z(1)_−x(1)_−w(1)_,y(2)_+z(2)_−x(2)_−w(2)_,..., y(n)_+z(n)_−x(n)_−w(n)_|x(1)_,x(2)_,...,x(n)_∈X(α),

y(2)_,y(2)_,...,y(n)_{∈Y (α),z}(1)_,z(2)_,...,z(n)_∈Z(α),

w(1),w(2),...,w(n)∈W (α).

(4.21)

For eachα∈[0,1], let

D=x(1),x(2),...,x(n)∈X(α), y(1),y(2),...,y(n)∈Y (α),

z(1)_,z(2)_,...,z(n)_∈Z(α),w(1)_,w(2)_,...,w(n)_{∈W (α).} (4.22)

FromRemark 4.4,

n

j=1

aj+bj2≤

n

j=1

a2

j+ n

j=1

b2

and inequality sup(A+B)≤supA+supB, we have

d∗Y (α)(+)Z(α)(−)X(α)(+)W (α)

=sup

D    n

j=1

y(j)_+z(j)_−x(j)_−w(j)2   

1/2

≤ sup

(x(j)_)∈X(α),(y(j)_{)∈Y (α)}

n

j=1

y(j)−x(j)2

+ sup

(w(j))∈W (α),(z(j))∈Z(α) n

j=1

z(j)−w(j)2

=d∗Y (α)(−)X(α)+d∗Z(α)(−)W (α).

(4.24)

ByDefinition 4.1, we have|X→Y⊕W→Z|∗_{≤ |X}→_Y|∗_+|W→_Z|∗_.

4.2. The fuzzy inner product and the angle between fuzzy vectors for the fuzzy vectors inSFR. Corresponding to the equation

dY (α)(−)X(α),V (α)(−)U(α)

= sup

(x(j)_)∈X(α),(y(j)_{)∈Y (α),(u}(j)_)∈U(α),(v(j)_{)∈V (α)}

n

j=1

y(j)_−x(j)_v(j)_−u(j)_,

(4.25)

we define the fuzzy inner product as follows.

Definition4.6. For →

XY , U→V∈SFR, define the fuzzy inner product of them to be

→

XY∗_U→_{V= sup} 0≤α≤1d

_{Y (α)(−)X(α),V (α)(−)U(α). (4.26)}

Property4.7. For →

XY , U→V∈SFR, let the 0-cuts(α-cuts, α=0)ofX,Y,U, andV beX(0),Y (0),U(0), andV (0), respectively. Then there exist

x(1)

m (0),x(m2)(0),...,x(n)m (0)

∈X(0),

y(1)

m (0),ym(2)(0),...,ym(n)(0)

∈Y (0),

u(1)

m(0),u(m2)(0),...,u(n)m (0)

∈U(0),

vm(1)(0),vm(2)(0),...,vm(n)(0)

∈V (0),

(4.27)

such that

→

XY∗_U→_V=n

j=1

EXTENSION OFn-DIMENSIONAL EUCLIDEAN VECTOR SPACEE ... 2365

Proof. _{Use the same way of the proof ofProperty 4.2to prove it.}

Property_4.8. _For →

XY , U→V , W→Z∈SFR,k1∈Fp(1 1),k >0,

(1) X→Y∗_U→_V=U→_V∗_X→_Y;

(2) X→Y∗_(U→_V⊕W→_{Z) ≤(X}→_Y∗_U→_{V ) ⊕(X}→_Y∗_W→_{Z) ;}

(3) k(X→Y∗_U→_{V ) =(k1X}→_{Y )} ∗_U→_V=X→_Y∗_(k1U→_{V ) ;}

(4) X→Y∗X→Y= |X→Y|∗2_;

(5) |X→Y∗_U→_{V| ≤ |X}→_Y|∗_·|U→_V|∗_.

Proof. _{(1) ByProperty 4.7,}

→

XY∗U→V= n

j=1

ym(j)(0)−xm(j)(0)vm(j)(0)−u(j)m(0)

=

n

j=1

vm(j)(0)−u(j)m(0)ym(j)(0)−xm(j)(0)=

→

UV∗_X→_{Y .}

(4.29)

(2) ByDefinition 4.6andProperty 3.6(1),U→V⊕W→Z=A→B=BA, whereA=

U⊕W, B=V⊕Z. Theα-cut ofBAis(V (α)(+)Z(α)(−)(U(α)(+)W (α)). Hence for eachα∈[0,1], set

H=x(1),x(2),...,x(n)∈X(α), y(1),y(2),...,y(n)∈Y (α),

u(1),u(2),...,u(n)∈U(α),v(1),v(2),...,v(n)∈V (α),

w(1),w(2),...,w(n)∈W (α), z(1),z(2),...,z(n)∈Z(α),

E=x(1)_,x(2)_,...,x(n)_{∈X(α), y}(1)_,y(2)_,...,y(n)_{∈Y (α),}

u(1)_,u(2)_,...,u(n)_∈U(α),v(1)_,v(2)_,...,v(n)_{∈V (α),}

D=x(1)_,x(2)_,...,x(n)_{∈X(α), y}(1)_,y(2)_,...,y(n)_{∈Y (α),}

w(1)_,w(2)_,...,w(n)_{∈W (α), z}(1)_,z(2)_,...,z(n)_∈Z(α).

(4.30)

Then, for eachα∈[0,1],

dY (α)(−)X(α),V (α)(+)Z(α)(−)U(α)(+)W (α)

=sup

H

j=1

y(j)−x(j)v(j)+z(j)−u(j)−w(j)

≤sup

E n

j=1

y(j)_−x(j)_v(j)_−u(j)

+sup

D n

j=1

y(j)−x(j)z(j)−w(j)

=dY (α)(−)X(α),V (α)(−)U(α)

+dY (α)(−)X(α),Z(α)(−)W (α).

ByDefinition 4.6, (2) is proved.

(3) The proof follows fromProperty 3.6(2) and (v), (vi). (4) The proof follows from Properties4.2and4.7. (5) For eachα, where 0≤α≤1,

x(1)_,x(2)_,...,x(n)_{∈X(α), y}(1)_,y(2)_,...,y(n)_{∈Y (α),}

u(1)_,u(2)_,...,u(n)_{∈U(α), v}(1)_,v(2)_,...,v(n)_{∈V (α),} (4.32)

by Cauchy-Schwartz inequality, that is,

−

n

j=1

y(j)−x(j)2 n

j=1

v(j)−u(j)2

≤

n

j=1

y(j)_−x(j)_v(j)_−u(j)

≤

n

j=1

y(j)_−x(j)2 n

j=1

v(j)_−u(j)2

(4.33)

since supAB≤supAsupB, ifA >0,B >0. Then, we have

−d∗Y (α)(−)X(α)d∗V (α)(−)U(α)

≤dY (α)(−)X(α),V (α)(−)U(α)

≤d∗Y (α)(−)X(α)d∗V (α)(−)U(α), ∀α∈[0,1].

(4.34)

Therefore, −|X→Y|∗_{· |U}→_V|∗ _{≤ |X}→_Y∗_U→_{V| ≤ |X}→_Y|∗_{· |U}→_V|∗_{. Hence, |X}→_Y∗ →

UV| ≤ |X→Y|∗_·|U→_V|∗_.

Remark_4.9. _If| →

XY|∗_{>0 and|U}→_V|∗_{>0, byProperty 4.8(5),}

−1≤ →

XY∗_U→_V

X→Y∗·U→V∗≤

1. (4.35)

So we have the following definition.

Definition4.10. For →

XY , U→V ∈SFR, ifX→Y=OO→,U→V =O→O, define the angleθbetweenX→YandU→V by

cosθ= →

XY∗_U→_V

_X→_Y∗·_U→_V∗.

(4.36)

Example4.11 (n=2). Eject the rocket from (2,3) aiming at (6,8). The

EXTENSION OFn-DIMENSIONAL EUCLIDEAN VECTOR SPACEE ... 2367

rocket aiming at(10,4), the rocket falls in the circle centered at(10,4)with radius 1.

Then we have the membership functions of the fuzzy setsX,Y,U, andV:

µX(2,3)=

  

1, ifx(1)_{=2, x}(2)_=3,

0, elsewhere,

µY

y(1),y(2)

=

    

1 4

4−y(1)₋₆2_−y(2)₋₈2_{, ify}(1)₋₆2_−y(2)₋₈2_≤4,

0, elsewhere,

µ_U(4,1)=

  

1, ifu(1)_{=4, u}(2)_=1,

0, elsewhere,

µ_Vv(1)_,v(2)₌   

1−v(1)₋₁₀2_−v(2)₋₄2_{, ifv}(1)₋₁₀2_−v(2)₋₄2_≤1,

0, elsewhere.

(4.37)

Theα-cuts, 0≤α≤1, ofX,Y,U,V are

X(α)=(2,3),

Y (α)=y(1)_,y(2)_|y(1)₋₆2_−y(2)₋₈2_{≤4(1−α),} U(α)=(4,1),

V (α)=v(1)_,v(2)_|v(1)₋₁₀2

−v(2)₋₄2

≤1−α, X(0)=(2,3),

Y (0)=y(1),y(2)|y(1)−62−y(2)−82≤4, U(0)=(4,1),

V (0)=v(1),v(2)|v(1)−102−v(2)−42≤1.

(4.38)

ByFigure 4.1, we have

x(m1)(0)=2, x(m2)(0)=3,

ym(1)(0)−2

4 =

√

41+2

√

41 , y

(1)

m (0)=6+

8

√

41,

ym(2)(0)−3

5 =

√

41+2

√

41 , y

(2)

m (0)=8+

10

√

41.

(4.39)

ByProperty 4.2, the length is

|X→Y|∗_=y(1)

15 10

5 0

5 10

v(1)₋102+v(2)₋42=1

(10,4)

V (0) vm(1)(0),vm(2)(0)

√ 45

U(0)=(4,1) X(0)=(2,3)

√ 41

Y (0) (6,8)

ym(1)(0),ym(2)(0)

y(1)−62+y(2)−82=4

Figure_4.1. _{Fuzzy vectors}X→YandU→V.

Similarly, we have

u(1)

m(0)=4, um(2)(0)=1,

v(1)

m (0)=10+

6

√

45, v

(2)

m (0)=4+

3

√

45,

(4.41)

and the length

_U→_V∗=vm(1)(0)−u(m1)(0)2+vm(2)(0)−um(2)(0)2=7.708. (4.42)

ByProperty 4.7,

→

XY∗_U→_V=y(1)

m (0)−xm(1)(0)

vm(1)(0)−u(m1)(0)

+ym(2)(0)−xm(2)(0)

vm(2)(0)−u(m2)(0)

=58.81125.

(4.43)

ByDefinition 4.10, the angle betweenX→YandU→Vhas

cosθ= →

XY∗_U→_V

_X→_Y∗·_U→_V∗=

0.907996. (4.44)

In crisp case, the vectorX→YfromX=(2,3)toY =(6,8)is(4,5)and the vector

→

EXTENSION OFn-DIMENSIONAL EUCLIDEAN VECTOR SPACEE ... 2369

|U→V| =6.708;X→Y·U→V=4·6+5·3=39 and

cosθ= →

XY·U→V

X→Y·U→V=

0.908005. (4.45)

Example4.12. Let

µX

x(1),x(2)

=

  

1−x(1)₋₅2_−x(2)₋₁₀2_{, ifx}(1)₋₅2_+x(2)₋₁₀2_≤1,

0, elsewhere,

µ_Yy(1)_,y(2)

=

    

1 4

4−y(1)₋₁₄2

−y(2)₋₁₅2

, ify(1)₋₁₄2

+y(2)₋₁₅2 ≤4,

0, elsewhere,

µ_Uu(1)_,u(2)

=

    

1 4

4−u(1)₋₈2_−u(2)₋₂2_{, ifu}(1)₋₈2_+u(2)₋₂2_≤4,

0, elsewhere,

µ_Vv(1)_,v(2)

=

  

1−v(1)₋₁₇2_−v(2)₋₇2_{, ifv}(1)₋₁₇2_+v(2)₋₇2_≤1,

0, elsewhere.

(4.46)

We have theα=0-cuts ofX,Y,U,V:

X(0)=x(1)_,x(2)_|x(1)₋₅2

+x(2)₋₁₀2 ≤1,

Y (0)=y(1)_,y(2)_|y(1)₋₁₄2_+y(2)₋₁₅2_≤4,

U(0)=u(1)_,u(2)_|u(1)₋₈2_+(u(2)₋₂2_≤4,

V (0)=v(1),v(2)|v(1)−172+v(2)−72≤1.

(4.47)

As inExample 4.11, fromFigure 4.2, we have

xm(1)(0)=5−

9

√

106, x

(2)

m(0)=10−

5

√

106,

ym(1)(0)=14+

18

√

106, y

(2)

m (0)=15+

10

√

106,

u(1)

m(0)=8−

18

√

106, u

(2)

m(0)=2−

10

√

106,

vm(1)(0)=17+

9

√

106, v

(2)

m (0)=7+

5

√

106.