Because, domain of fuzzy attributes can except values are number, linguistic values, thus we have to effect and simply on method to approximate data.. T´ om t˘ a ´t.[r]
(1)Ta.p ch´ı Tin ho.c v`a Diˆe` u khiˆe’n ho.c, T.23, S.2 (2007), 110–121
M ˆO T CACH TIˆ´ E´P CˆA N DEˆ’ X ˆA´P XI’ D˜U LIˆE U TRONG CO.SO’ D˜ U LIˆE U MO`
NGUY ˆE˜ N C´AT H ˆO`1, NGUY ˆE˜ N C ˆONG H`AO2
Viˆe.n Cˆong nghˆe thˆong tin, Viˆe.n Khoa ho.c v`a Cˆong nghˆe Viˆe.t Nam
2
Tru.`o.ng Da.i ho.c Khoa ho.c Huˆe´
Abstract In this paper, we introduced a method to approximate data on domain of fuzzy attributes in relation of fuzzy databases based hedge algebra Because, domain of fuzzy attributes can except values are number, linguistic values, thus we have to effect and simply on method to approximate data
T´om t˘a´t B`ai b´ao tr`ınh b`ay mˆo.t phu.o.ng ph´ap xˆa´p xı’ d˜u liˆe.u trˆen miˆe` n tri thuˆo.c t´ınh m`o cu’a mˆo.t quan hˆe co so.’ d˜u liˆe.u m`o du a trˆen da.i sˆo´ gia tu.’ Bo.’i v`ı miˆe` n tri cu’a thuˆo.c t´ınh m`o c´o thˆe’ l`a gi´a tri sˆo´, gi´a tri ngˆon ng˜u., d´o ch´ung ta cˆa` n c´o mˆo.t phu.o.ng ph´ap xˆa´p xı’ d˜u liˆe.u mˆo.t c´ach do.n gia’n v`a hiˆe.u qua’.
1 D ˘A T VAˆ´N Dˆ`E
Co so.’ d˜u liˆe.u m`o d˜a du.o c nhiˆe`u t´ac gia’ v`a ngo`ai nu.´o.c quan tˆam nghiˆen c´u.u v`a d˜a c´o nh˜u.ng kˆe´t qua’ d´ang kˆe’ ([1–5, 10, 12]) C´o nhiˆe` u c´ach tiˆe´p cˆa.n kh´ac nhu c´ach tiˆe´p cˆa.n theo l´y thuyˆe´t tˆa.p m`o ([1]), theo l´y thuyˆe´t kha’ n˘ang ([4]) Prade v`a Testemale n˘am 1983, quan hˆe tu.o.ng du.o.ng ([2, 3, 5]) Tˆa´t ca’ c´ac c´ach tiˆe´p cˆa.n trˆen nh˘a`m mu.c d´ıch n˘a´m b˘a´t v`a xu.’ l´y mˆo.t c´ach tho’a d´ang trˆen mˆo.t luˆa.n diˆe’m n`ao d´o c´ac thˆong tin khˆong ch´ınh x´ac (unexact), khˆong ch˘a´c ch˘a´n (uncertainty) hay nh˜u.ng thˆong tin khˆong dˆa` y du’ (incomplete) Do su da da.ng cu’a nh˜u.ng loa.i thˆong tin n`ay nˆen ta g˘a.p rˆa´t kh´o kh˘an biˆe’u thi ng˜u ngh˜ıa v`a thao t´ac v´o.i ch´ung
Trong th`o.i gian qua, da.i sˆo´ gia tu.’ du.o c nhiˆe` u t´ac gia’ nghiˆen c´u.u [6–8] v`a d˜a c´o nh˜u.ng ´u.ng du.ng d´ang kˆe’, d˘a.c biˆe.t lˆa.p luˆa.n xˆa´p xı’ v`a mˆo.t sˆo´ b`ai to´an diˆe` u khiˆe’n V`ı vˆa.y, viˆe.c nghiˆen c´u.u vˆe` co so.’ d˜u liˆe.u m`o theo c´ach tiˆe´p cˆa.n da.i sˆo´ gia tu.’ l`a mˆo.t hu.´o.ng m´o.i cˆa` n quan tˆam gia’i quyˆe´t
2 DA I SOˆ´ GIA TU.’
Dˆe’ xˆay du. ng c´ach tiˆe´p cˆa.n da.i sˆo´ gia tu’ , phˆa ` n n`ay s˜e tr`ınh b`ay tˆo’ng quan vˆe` mˆo.t sˆo´ n´et co ba’n cu’a da.i sˆo´ gia tu.’ v`a kha’ n˘ang biˆe’u thi ng˜u ngh˜ıa du a v`ao cˆa´u tr´uc cu’a da.i sˆo´ gia tu.’ , h`am di.nh lu.o ng ng˜u ngh˜ıa v`a mˆo.t sˆo´ t´ınh chˆa´t cu’a da.i sˆo´ gia tu.’.
Ta x´et miˆe` n ngˆon ng˜u cu’a biˆe´n chˆan l´y TRUTH gˆo`m c´ac t`u sau:
(2)M ˆO T C ´ACH TI ˆE´P CˆA.N DˆE’ XˆA´P XI’ D˜U.LIˆE U 111
possiblytrue,possibly false,approximatelytrue,approximatelyfalse, littletrue, littlefalse,very possibly true,very possibly false }, d´o true, false l`a c´ac t`u nguyˆen thuy’, c´ac t`u nhˆa´n (mordifier hay intensifier) very, more-or-less, possibly, approximately, little go.i l`a c´ac gia tu.’ (hedges) Khi d´o miˆe` n ngˆon ng˜u.T =dom(TRUTH) c´o thˆe’ biˆe’u thi nhu mˆo.t da.i sˆo´AH = (X, G, H,6), d´oGl`a tˆa.p c´ac t`u nguyˆen thuy’ du.o c xem l`a c´ac phˆa` n tu.’ sinh H l`a tˆa.p c´ac gia tu.’ du.o c xem nhu l`a c´ac ph´ep to´an mˆo.t ngˆoi, quan hˆe (trˆen c´ac t`u (c´ac kh´ai niˆe.m m`o.) l`a quan hˆe th´u tu du.o c “ca’m sinh” t`u ng˜u ngh˜ıa tu nhiˆen V´ı du du a trˆen ng˜u ngh˜ıa, c´ac quan hˆe th´u tu sau l`a d´ung: false6true,moretrue6verytrue nhu.ngveryfalse6morefalse, possibly true true nhu.ng false possibly false Tˆa.p X du.o. c sinh t`u G bo.’ i c´ac ph´ep t´ınh H Nhu vˆa.y mˆo˜i phˆa` n tu.’ cu’aX s˜e c´o da.ng biˆe’u diˆe˜n x=hnhn−1 h1x, x∈G
Tˆa.p tˆa´t ca’ c´ac phˆa` n tu.’ du.o c sinh t`u mˆo.t phˆa`n tu.’ x du.o c k´y hiˆe.u l`aH(x) Nˆe´uGc´o d´ung hai t`u nguyˆen thuy’ m`o., th`ı mˆo.t du.o c go.i l`a phˆa` n tu.’ sinh du.o.ng k´y hiˆe.u l`a c+, mˆo.t go.i l`a phˆa` n tu.’ sinh ˆam k´y hiˆe.u l`a c− v`a ta c´o c− < c+ Trong v´ı du trˆen true l`a du.o.ng c`on false l`a ˆam Cho da.i sˆo´ gia tu.’X= (X, G, H,6), v´o.i G={c+, c−}, d´oc+ v`ac− tu.o.ng ´u.ng l`a phˆa` n tu.’ sinh du.o.ng v`a ˆam, X l`a tˆa.p nˆe` n H = H+∪H− v´o.i H− = {h1, h2, , hp} v`a
H+={h
p+1, , hp+q}, h1 > h2> > hp v`ahp+1< < hp+q
Di.nh ngh˜ıa 2.1 ([9]) f :X →[0,1]go.i l`a h`am di.nh lu.o ng ng˜u ngh˜ıa cu’aX nˆe´u∀h,∈H+ ho˘a.c ∀h, k∈H−v`a∀x, y∈X,ta c´o:
f(hx)−f(x) f(kx)−f(x)
=
f(hy)−f(y) f(ky)−f(y)
V´o.i da.i sˆo´ gia tu.’ v`a h`am di.nh lu.o ng ng˜u ngh˜ıa ta c´o thˆe’ di.nh ngh˜ıa t´ınh m`o cu’a mˆo.t kh´ai niˆe.m m`o Cho tru.´o.c h`am di.nh lu.o ng ng˜u ngh˜ıaf cu’a X X´et bˆa´t k`y x∈X.T´ınh m`o cu’a xkhi d´o du.o. c b˘a`ng du.`o.ng k´ınh cu’a tˆa.pf(H(x))⊆[0,1]
H`ınh T´ınh m`o cu’a gi´a tri True
Di.nh ngh˜ıa 2.2 [9] H`amf m:X→[0,1]du.o c go.i l`adˆo t´ınh m`o. trˆen X nˆe´u thoa’ m˜an c´ac diˆe` u kiˆe.n sau:
(1) f m(c−) =W >0 v`af m(c+) = 1−W >0 (2) V´o.i c∈ {c−, c+} th`ı
p+q P i=1
f m(hic) =f m(c)
(3) V´o.i mo.i x, y∈X,∀h∈H, f m(hx) f m(x) =
f m(hy) f m(y) =
f m(hc)
f m(c) , v´o.i c∈ {c
(3)112 NGUYˆE˜ N C´AT HˆO`, NGUYˆE˜N CˆONG H`AO
ngh˜ıa l`a tı’ sˆo´ n`ay khˆong phu thuˆo.c v`ao x v`a y, du.o c k´ı hiˆe.u l`a µ(h) go.i l`a dˆo t´ınh m`o. (fuzziness measure) cu’a gia tu.’ h
Mˆe.nh dˆe` 2.1 [9]
(1) f m(hx) =µ(h)f m(x), v´o.i mo.i x∈X (2)
p+q P i=1
f m(hic) =f m(c), d´o c∈ {c−, c+}
(3)
p+q P i=1
f m(hix) =f m(x), ∀x∈X
(4)
p P i=1
µ(hi) =α v`a p+q
P i=p+1
µ(hi) =β, v´o.iα, β >0 v`aα+β =
Di.nh ngh˜ıa 2.3 [9] H`amSign:X → {−1,0,1}l`a mˆo.t ´anh xa du.o c di.nh ngh˜ıa mˆo.t c´ach
dˆe qui nhu sau, v´o.i mo.i h, h0∈H:
(1) Sign(c−) =−1 v`aSign(hc−) = +Sign(c−)nˆe´uhc−< c− Sign(hc−) =−Sign(c−) nˆe´uhc−> c−
Sign(c+) = +1v`aSign(hc+) = +Sign(c+) nˆe´uhc+ > c+
Sign(hc+) =−Sign(c+) nˆe´uhc+< c+
(2) Sign(h0hx) =−Sign(hx)nˆe´uh0 l`a negative dˆo´i v´o.ihv`ah0hx6=hx (3) Sign(h0hx) = +Sign(hx)nˆe´uh0 l`a positive dˆo´i v´o.ihv`ah0hx6=hx (4) Sign(h0hx) = 0nˆe´uh0hx=hx
Di.nh ngh˜ıa 2.4 [9] Gia’ su.’ cho tru.´o.c dˆo t´ınh m`o cu’a c´ac gia tu.’µ(h), v`a c´ac gi´a tri. dˆo t´ınh m`o cu’a c´ac phˆa` n tu.’ sinh f m(c−), f m(c+) v`a wl`a phˆa` n tu.’ trung h`oa H`am di.nh
lu.o ng ng˜u ngh˜ıa (quantitatively semantic function) ν cu’a X du.o. c xˆay du ng nhu sau v´o.i x=him hi2hi1c:
(1) ν(c−) =W −α.f m(c−) v`aν(c+) =W +α.f m(c+) (2) ν(hjx) =
ν(x)+Sign(hjx)× hXp
i=j
f m(hix)−
1−Sign(hjx)Sign(h1hjx)(β−α)
f m(hjx) i
v´o.i 16j6p,v`a
ν(hjx) =ν(x)+Sign(hjx)× h Xj
i=p+1
f m(hix)−
1−Sign(hjx)Sign(h1hjx)(β−α)
f m(hjx) i
v´o.i j > p
3 M ˆO T CACH TIˆ´ E´P C ˆA N DEˆ’ X ˆA´P XI’ D ˜U.LIˆE U MO`
Trong mu.c n`ay, s˜e tr`ınh b`ay mˆo.t phu.o.ng ph´ap m´o.i dˆe’ xˆa´p xı’ d˜u liˆe.u trˆen miˆe` n tri cu’a thuˆo.c t´ınh m`o quan hˆe cu’a co so.’ d˜u liˆe.u m`o Viˆe.c d´anh gi´a d˜u liˆe.u trˆen miˆe` n tri thuˆo.c t´ınh m`o cu’a quan hˆe co so.’ d˜u liˆe.u m`o theo c´ach tiˆe´p cˆa.n da.i sˆo´ gia tu.’ du.o c xˆay du ng du. a trˆen phˆan hoa.ch t´ınh m`o cu’a c´ac gi´a tri da.i sˆo´ gia tu.’ (gi´a tri ngˆon ng˜u.) Nhu vˆa.y, nˆe´u go.i Dom(Ai) l`a miˆe` n tri tu.o.ng ´u.ng v´o.i thuˆo.c t´ınh m`o.Ai v`a xem nhu mˆo.t da.i sˆo´ gia
tu.’ th`ı d´o Dom(Ai) = Num(Ai)∪LV(Ai), v´o.i Num(Ai) l`a tˆa.p c´ac gi´a tri sˆo´ cu’a Ai v`a
(4)M ˆO T C ´ACH TI ˆE´P CˆA.N DˆE’ XˆA´P XI’ D˜U.LIˆE U 113 3.1 Miˆ` n tri cu’a thuˆo.c t´ınh quan hˆe l`a gi´a tri ngˆon ng˜ue
Trong tru.`o.ng ho p n`ay, ch´ung ta di xˆay du ng c´ac phˆan hoa.ch du a v`ao t´ınh m`o cu’a c´ac gi´a tri ngˆon ng˜u
V`ı t´ınh m`o cu’a c´ac gi´a tri da.i sˆo´ gia tu.’ l`a mˆo.t doa.n cu’a [0,1] cho nˆen ho c´ac doa.n nhu vˆa.y cu’a c´ac gi´a tri c´o c`ung dˆo d`ai s˜e ta.o th`anh phˆan hoa.ch cu’a [0,1] Phˆan hoa.ch ´u.ng v´o.i c´ac gi´a tri c´o dˆo d`ai t`u l´o.n ho.n s˜e mi.n ho.n v`a dˆo d`ai l´o.n vˆo ha.n th`ı dˆo d`ai cu’a c´ac doa.n phˆan hoa.ch gia’m dˆa` n vˆe`
Di.nh ngh˜ıa 3.1 Go.i f m l`a dˆo t´ınh m`o trˆen DSGT X V´o.i mˆo˜i x ∈ X, ta k´y hiˆe.u I(x)⊆[0,1]v`a|I(x)|l`a dˆo d`ai cu’aI(x)
Mˆo.t ho c´ac ξ={I(x) :x∈X} du.o. c go.i l`a phˆan hoa.ch cu’a [0,1] g˘a´n v´o.ixnˆe´u: (1) {I(c+), I(c−)}l`a phˆan hoa.ch cu’a [0,1] cho|I(c)|=f m(c), v´o.ic∈ {c+, c−}.
(2) Nˆe´u doa.n I(x)d˜a du.o c di.nh ngh˜ıa v`a|I(x)|=f m(x)th`ı{I(hix) :i= p+q}du.o c
di.nh ngh˜ıa l`a phˆan hoa.ch cu’aI(x)sao cho thoa’ m˜an diˆe` u kiˆe.n|I(hix)|=f m(hix)v`a|I(hix)|
l`a tˆa.p s˘a´p th´u tu tuyˆe´n t´ınh
Tˆa.p {I(hix)} du.o c go.i l`a phˆan hoa.ch g˘a´n v´o.i phˆa` n tu.’x Ta c´o p+q
P i=1
|I(hix)|= |I(x)|=
f m(x)
Di.nh ngh˜ıa 3.2 ChoPk ={I(x) :x∈XXXk}v´o.iXXXk={x∈XXX:|x|=k}
l`a mˆo.t phˆan hoa.ch Ta n´oi r˘a`nguxˆa´p xı’ν theo m´u.cktrong Pk du.o. c k´y hiˆe.uu≈kν v`a chı’ khiI(u) v`aI(v)
c`ung thuˆo.c mˆo.t khoa’ng trong Pk C´o ngh˜ıa l`a ∀u, v ∈XXX, u≈k v ⇔ ∃∆k∈Pk :I(u)⊆∆k
v`aI(v)⊆∆k.
V´ı du 3.1. Cho da.i sˆo´ gia tu.’ X = (XXX, G, H,6),trong d´o H = H+∪H−, H+ = {ho.n,
rˆa´t}, ho.n <rˆa´t, H−={´ıt, kha’ n˘ang}, ´ıt> kha’ n˘ang,G={ tre’, gi`a} Ta c´oP1 ={I(tre’),
I(gi`a)} l`a mˆo.t phˆan hoa.ch cu’a[0,1] Tu.o.ng tu. ,P2 ={I(ho.n tre’),I(rˆa´t tre’),I(´ıt tre’),I(kha’ n˘ang tre’), I(ho.n gi`a),I(rˆa´t gi`a),I(´ıt gi`a),I(kha’ n˘ang gi`a)} l`a phˆan hoa.ch cu’a[0,1]
V´ı du 3.2. Theo V´ı du 3.1, P1 l`a phˆan hoa.ch cu’a [0,1] Ta c´o ho.n tre’ ≈1 rˆa´t tre’ v`ı
∃∆1 = I(tre’)∈ P1 m`a I(ho.n tre’) ⊆∆1 v`a I(rˆa´t tre’) ⊆∆1.P2 l`a phˆan hoa.ch cu’a [0,1], ta c´o´ıt gi`a ≈2 rˆa´t ´ıt gi`a v`ı∃∆2 =I(´ıt gi`a )∈P2 m`aI(´ıt gi`a)⊆∆2 v`aI(rˆa´t ´ıt gi`a)⊆∆2 Di.nh ngh˜ıa 3.3 X´etPk ={I(x) :x ∈XXXk}v´o.iXXXk ={x∈XXX :|x|=k}l`a mˆo.t phˆan hoa.ch.
Ta n´oi r˘a`ng u khˆong xˆa´p xı’ v m´u.c k Pk du.o.
c k´y hiˆe.u u 6=k v v`a chı’ khiI(u) v`a
I(v)khˆong c`ung thuˆo.c mˆo.t khoa’ng trong Pk C´o ngh˜ıa l`a∀u, v∈XXX, u6=
k v⇔ ∀∆k∈Pk :
I(u)6⊂∆k
ho˘a.c I(v)6⊂∆k.
V´ı du 3.3. Theo V´ı du 3.1,P2 ={I(ho.n tre’), I(rˆa´t tre’), I(´ıt tre’),I(kha’ n˘ang tre’),I(ho.n gi`a), I(rˆa´t gi`a), I(´ıt gi`a), I(kha’ n˘ang gi`a)} l`a phˆan hoa.ch cu’a [0,1] Cho.n ∆2 = I(rˆa´t tre’)∈ P2, ta c´o I(´ıt tre’) 6⊂ ∆2 v`a I(rˆa´t tre’) ⊆ ∆2 (1’) M˘a.c kh´ac v´o.i mo.i ∆2 6= I(´ıt tre’)
∈P2 ta c´o I(´ıt tre’)6⊂∆2 v`aI(rˆa´t tre’) 6⊂∆2 (2’) T`u (1’) v`a (2’) ta suy ra´ıt tre’ 6=2 rˆa´t tre’ Di.nh ngh˜ıa 3.4 X´et Pk = {I(x) : x ∈ XXXk} v´o.i XXXk = {x ∈ XXX : |x| = k}
l`a mˆo.t phˆan hoa.ch Go.iν l`a h`am di.nh lu.o ng ng˜u ngh˜ıa trˆenXXX Ta n´oi r˘a`ngunho’ ho.nv m´u.cktrongPk
(5)114 NGUYˆE˜ N C´AT HˆO`, NGUYˆE˜N CˆONG H`AO
ν(u)< u(v) C´o ngh˜ıa l`a ∀u, v∈XXX, u <k v⇔u6=kv v`aν(u)< v(v)
V´ı du 3.4. Theo V´ı du 3.1 v`a 3.3 ta c´oP2 ={I(ho.n tre’), I(rˆa´t tre’), I(´ıt tre’), I(kha’ n˘ang tre’),I(ho.n gi`a),I(rˆa´t gi`a), I(´ıt gi`a), I(kha’ n˘ang gi`a)}l`a phˆan hoa.ch cu’a[0,1].V`ı´ıt tre’ 6=2
rˆa´t tre’ v`av(rˆa´t tre’ ) < v (´ıt tre’)nˆenrˆa´t tre’ <2´ıt tre’
C´ac di.nh l´y, hˆe qua’ v`a bˆo’ dˆe` liˆen quan dˆe´n nh˜u.ng quan hˆe du.o c dˆe` xuˆa´t Mu.c 3.1 nhu xˆa´p xı’, khˆong xˆa´p xı’ theo m´u.c phˆan hoa.ch s˜e du.o c tr`ınh b`ay v`a ch´u.ng minh dˆa` y du’ l`am co so.’ cho c´ac phˆa` n tiˆe´p theo
Bˆo’ dˆ` 3.1.e Quan hˆe. ≈k l`a mˆo.t quan hˆe tu.o.ng du.o.ng trˆen Dom(Ai)
Ch´u.ng minh: Ta ch´u.ng minh t´ınh pha’n xa b˘a`ng quy na.p.
∀x ∈Dom(Ai) nˆe´u|x|= 1th`ıx=c+ ho˘a.cx=c−
Ta c´o ∃∆1 =I(c+) ∈P1 :I(c+) = I(x)⊆∆1 ho˘a.c∃∆1 =I(c−)∈P1 :I(c−) = I(x)⊆
∆1.Vˆa.y≈k d´ung v´o.ik= 1,hayx ≈1 x
Gia’ su.’ |x| = n d´ung, c´o ngh˜ıa ≈k d´ung v´o.i k = n, hay x ≈n x, ta cˆa` n ch´u.ng minh
≈k d´ung v´o.i k = n+ D˘a.t x = h1x0, v´o.i |x0| =n V`ıx ≈n x nˆen theo di.nh ngh˜ıa ta c´o
∃∆n ∈ Pn : I(x) ⊆∆n.
M˘a.c kh´ac ta c´o Pn+1 ={I(h1x0), I(h2x0), }, v´o.ih1 6=h2 6= l`a
mˆo.t phˆan hoa.ch cu’a I(x0) Do d´o ∃∆(n+1) = I(h1x0) ∈ P(n+1) : I(h1x0) = I(x) ⊆ ∆(n+1)
Vˆa.y≈k d´ung v´o.i k=n+ 1, hay x≈n+1 x
T´ınh dˆo´i x´u.ng: ∀x, y∈Dom(Ai), nˆe´ux≈ky th`ı theo di.nh ngh˜ıa ∃∆k∈Pk :I(x)⊆∆k
v`aI(y)⊆∆k hay ∃∆k ∈Pk:I(y)⊆∆k v`aI(x)⊆∆k
Vˆa.y y≈k xth`ıy≈k x
T´ınh b˘a´t cˆa` u: Ta ch´u.ng minh b˘a`ng phu.o.ng ph´ap qui na.p. Tru.`o.ng ho p k= 1:
Ta c´o P1 ={I(c+), I(c−)}, nˆe´ux ≈1 y v`a y≈1 z th`ı∃∆1 =I(c+) ∈P1 :I(x)⊆∆1 v`a
I(y)⊆∆1 v`aI(z)⊆∆1 ho˘a.c∃∆1 =I(c−)∈P1 :I(x)⊆∆1 v`aI(y)⊆∆1 v`aI(z)⊆∆1, c´o ngh˜ıa l`a ∃∆1∈P1 :I(x)⊆∆1 v`aI(z)⊆∆1 hayx≈1 z.Vˆa.y ≈k d´ung v´o.ik=
Gia’ su.’ quan hˆe.≈k d´ung v´o.i tru.`o.ng ho p k=nc´o ngh˜ıa l`a ta c´o ∀x, y, z∈Dom(Ai)nˆe´u
x≈ny v`ay≈nz th`ıx≈nz
Ta cˆa` n ch´u.ng minh quan hˆe.≈kd´ung v´o.i tru.`o.ng ho pk=n+1.T´u.c l`a∀x, y, z∈Dom(Ai)
nˆe´ux≈n+1 y v`ay≈n+1 zth`ıx≈n+1 z
Theo gia’ thiˆe´t nˆe´ux≈n+1 yv`ay≈n+1 zth`ı∃∆(n+1) ∈P(n+1) :I(x)⊆∆(n+1)v`aI(y)⊆ ∆(n+1) v`a I(z) ⊆ ∆(n+1), c´o ngh˜ıa l`a ∃∆(n+1) ∈P(n+1) :I(x) ⊆∆(n+1) v`a I(z) ⊆∆(n+1) Vˆa.yx≈n+1 z
Bˆo’ dˆ` 3.2.e Cho u=hn h1x v`av=h0m h01x l`a biˆe’u diˆe˜n ch´ınh t˘a´c cu’a u v`av dˆo´i v´o.ix (1) Nˆe´uu=v th`ıu≈k v v´o.i mo.i k
(2) Nˆe´uh1 6=h01 th`ıu≈|x|v
Ch´u.ng minh:
(1) Theo Bˆo’ dˆe` 3.1, v`ıu=v nˆen ta c´ou≈ku hayv≈kv , v´o.i mo.i k
(2) Nˆe´u u| = |v| = 2, t´u.c l`a u = h1x v`a v = h01x, h1 =6 h01 nˆen u 6= v Ta c´o
I(h1x)⊆ I(x), I(h10x) ⊆I(x) v`aI(h1x) 6⊂I(h01x) nˆen ∃∆1 = I(x) ∈P1 :I(h1x) ⊆ ∆1 v`a
I(h01x)⊆∆1 hay h1x≈1 h01x.Vˆa.yu≈|x|v
(6)M ˆO T C ´ACH TI ˆE´P CˆA.N DˆE’ XˆA´P XI’ D˜U.LIˆE U 115
∃∆k∈Pk ={I(h
k−1 h1x), I(h0k−1 h01x)}, v´o.iPk l`a mˆo.t phˆan hoa.ch cu’aI(x) :I(u)⊆∆k
v`aI(v)⊆∆k.
Nˆe´u cho.n ∆k = I(h
k−1 h1x) th`ı I(u) ⊆ I(hk−1 h1x) v`a I(v) ⊆ I(hk−1 h1x) hay
I(hn h1x)⊆I(hk−1 h1x)v`aI(h0m h10x)⊆I(hk−1 h1x)diˆe` u n`ay mˆau thuˆa’n v`ıI(h0m h01x)6⊂
I(hk−1 h1x)do (1’)
Nˆe´u cho.n∆k=I(h0
k−1 h01x)th`ıI(hn h1x)⊆I(h0k−1 h10x)v`aI(h0m h01x)⊆I(h0k−1 h01x),
diˆe` u n`ay mˆau thuˆa’n v`ıI(hn h1x)6⊂I(h0k−1 h01x) (1’) Vˆa.y khˆong tˆo`n ta.ik >1sao cho
u≈kv hayk= 1.Vˆa.yu≈|x|v
Di.nh l´y 3.1 X´et Pk ={I(x) : x ∈XXXk} v´o.iXXXk ={x ∈XXX : |x|=k}
l`a mˆo.t phˆan hoa.ch, u=hn h1x v`a v=h0m h01x l`a biˆe’u diˆe˜n ch´ınh t˘a´c cu’au v`av dˆo´i v´o.i x
(1) Nˆe´u u≈k v th`ıu≈k0 v, ∀0< k0< k
(2) Nˆe´u tˆo`n ta.i mˆo.t chı’ sˆo´ j min(m, n) l´o.n nhˆa´t cho v´o.i mo.i s = j, ta c´o hs=h0s th`ıu≈j+|x|v
Ch´u.ng minh: (1) Ta c´o Pk ={I(h
k−1 h1x), I(h0k−1 h1x)} V`ıu ≈k v nˆen theo di.nh ngh˜ıa
∃∆k∈Pk :I(u)⊆∆k v`aI(v)⊆∆k (1’).
Ta la.i c´o P1 = {I(x)}, P2 = {I(h
1x), I(h01x)}, , Pk ={I(hk−1 h1x), I(hk0−1 h1x)}
M˘a.t kh´ac ta c´o I(hk−1 h1x) ⊆ I(hk−2 h1x) ⊆ ⊆ I(h1x) ⊆ I(x) v`a I(h0k−1 h01x) ⊆
I(h0
k−2 h01x)⊆ ⊆I(h01x)⊆I(x)nˆen∃∆k=I(hk−1 h1x)∈Pkho˘a.c∃∆k=I(h0k−1 h01x)∈
Pk v`a ∃∆k−1 = I(hk−2 h1x) ∈ Pk−1 ho˘a.c ∃∆k−1 = I(h0k−2 h10x) ∈ Pk −1 v`a ∃∆2 =
I(h1x) ∈ P2 ho˘a.c ∃∆2 = I(h01x) ∈ P2 v`a ∃∆1 = I(x) ∈ P1 cho: ∆k ⊆ ∆k−1 ⊆ ⊆ ∆2⊆∆1 (2’)
T`u (1’) v`a (2’) ta c´o I(u) ⊆ ∆k ⊆ ∆k−1 ⊆ ⊆ ∆2 ⊆ ∆1 v`a I(v) ⊆ ∆k ⊆ ∆k−1 ⊆
⊆ ∆2 ⊆ ∆1, c´o ngh˜ıa l`a ∀0 < k0 < k luˆon ∃∆k0
∈ Pk0
:I(u) ⊆∆k0
v`aI(v) ⊆∆k0
.Vˆa.y
∀0< k0 < knˆe´uu≈kv th`ıu≈k0 v
(2): Nˆe´uj= 1ta c´oh1 =h01, d´ou=hn h2h1xv`av=h0m h02h01xhayu=hn h2h1x
v`av=h0m h02h1x.D˘a.t x0 =h1x ta c´ou=hn h2x0 v`av=h0m h20x0 V`ıh26=h02 nˆen theo
Bˆo’ dˆe` 2.3 ta c´ou≈|x0|v(do |x0|= 2, |x|= 1) hayu≈2 v Vˆa.yu≈j+|x|v
Nˆe´u j 6= 1, d˘a.t k=j, ta cˆa` n ch´u.ng minh u≈k+|x| v.V`ıu≈k v nˆen theo gia’ thiˆe´t ta c´o
∀s= kta c´ohs =h0s Khi d´o u=hn h2h1x v`av=h0m h02h01x hayu=hn.hkhk−1 h1x
v`av=h0m hkhk−1 h1x
D˘a.t x0 = hkhk−1 h1x ta c´o u =hn hk+1x0 v`a v = h0m h0k+1x0 V`ıhk+1 6= h0k+1 nˆen
theo Bˆo’ dˆe` 2.2 ta c´o u≈|x0|v hay u≈k+|x|v(do |x0|=k, |x|= 1)
Hˆe qua’ 3.1. Nˆe´u u∈H(v) th`ıu≈|v|v
Di.nh l´y 3.2 X´et Pk = {I(x) : x ∈XXXk} v´o.i XXXk = {x ∈ XXX : |x|= k}, u = hn h1x v`a
v=h0m h01x l`a biˆe’u diˆe˜n ch´ınh t˘a´c cu’a u v`a v dˆo´i v´o.i x Nˆe´u tˆo`n ta.i chı’ sˆo´k6min(m, n)
l´o.n nhˆa´t chou≈kv th`ıu6=k+1 v
Hˆe qua’ 3.2. (1) Nˆe´uu ∈H(v) th`ıu6=|v|+1 v
(2) Nˆe´u u6=kv th`ıu6=k0 v ∀0< k < k0
Di.nh l´y 3.3 X´et Pk = {I(x) : x ∈XXXk} v´o.i XXXk = {x ∈ XXX : |x|= k}, u = h
n h1x v`a
(7)116 NGUYˆE˜ N C´AT HˆO`, NGUYˆE˜N CˆONG H`AO
mo.ia∈H(u), v´o.i mo.i b∈H(v) ta c´oa <kb ho˘a.c a >kb
3.2 Miˆ` n tri cu’a thuˆo.c t´ınh quan hˆe c´o ch´u.a gi´a tri sˆo´e
Tru.`o.ng ho p miˆe` n tri cu’a thuˆo.c t´ınh c´o ch´u.a gi´a tri sˆo´, ch´ung ta s˜e biˆe´n dˆo’i c´ac gi´a tri sˆo´ th`anh c´ac gi´a tri ngˆon ng˜u tu.o.ng ´u.ng theo mˆo.t ng˜u ngh˜ıa x´ac di.nh Tru.´o.c tiˆen, ta di xˆay du. ng mˆo.t h`am IC chuyˆe’n mˆo.t sˆo´ vˆe` mˆo.t gi´a tri thuˆo.c [0,1] v`a h`amΦk dˆe’ chuyˆe’n mˆo.t gi´a
tri trong[0,1]th`anh mˆo.t gi´a tri ngˆon ng˜u.xtu.o.ng ´u.ng da.i sˆo´ gia tu.’XXX
Di.nh ngh˜ıa 3.5 ChoDom(Ai) =N um(Ai)∪LV(Ai), v l`a h`am di.nh lu.o ng ng˜u ngh˜ıa cu’a
Ai H`am IC :Dom(Ai)→[0,1]du.o c x´ac di.nh nhu sau:
Nˆe´u LV(Ai) = ∅ v`a N um(Ai) 6= ∅ th`ı∀ω ∈ Dom(Ai) ta c´o IC(ω) =
ω−ψmin
ψmax−ψmin
v´o.i Dom(Ai) = [ψmin, ψmax]l`a miˆe` n tri kinh diˆe’n cu’aAi
Nˆe´uN um(Ai)6=∅, LV(Ai)6=∅th`ı∀ω∈Dom(Ai)ta c´oIC(ω) ={ω∗v(ψmaxLV)}/ψmax,
v´o.i LV(Ai) = [ψminLV, ψmaxLV]l`a miˆe` n tri ngˆon ng˜u cu’aAi
V´ı du 3.5. ChoDom(T uoi) ={0 100, rˆa´t rˆa´t tre’, , rˆa´t rˆa´t gi`a} N um(T uoi) ={20,25,27,30,45,60,75,66,80}
LV(T uoi) ={tre’, rˆa´t tre’, gi`a, kh´a tre’, kh´a gi`a, ´ıt gi`a, rˆa´t gi`a, rˆa´t rˆa´t tre’}, Dom(T uoi) = N um(T uoi)∪LV(T uoi)
Nˆe´uLV(T uoi) =∅khi d´oDom(T uoi) =N um(T uoi) ={20,25,27,30,45,60,75,66,80} Do d´o ∀ω ∈ Dom(T uoi),ta c´o Dom(T uoi) = {0,2, 0,25, 0,27, 0,3, 0,45, 0,6, 0,75, 0,66, 0,8}
Nˆe´u N um(Ai)6=∅v`aLV(Ai)6=∅ ta c´oDom(T uoi) =N um(T uoi)∪LV(T uoi) ={tre’,
rˆa´t tre’, gi`a, kh´a tre’, kh´a gi`a, ´ıt gi´a, rˆa´t gi`a, rˆa´t rˆa´t tre’, 20, 25, 27, 30, 45, 60, 75, 66, 80} Gia’ su.’ t´ınh du.o c v(ψmaxLV) = v(rˆa´t rˆa´t gi`a) = 0,98 Khi d´o ∀ω ∈ N um(Ai) ta c´o
IC(ω) ={ω.v(ψmaxLV)}/ψmax= (ω×0,98)/100,hay∀ω ∈N um(Ai)su.’ du.ngIC(ω), ta c´o
N um(Ai) ={0,196, 0,245, 0,264, 0,294, 0,441, 0,588, 0,735, 0,646, 0,784}
Nˆe´u ta cho.n c´ac tham sˆo´W v`a dˆo t´ınh m`o cho c´ac gia tu.’ chov(ψmaxLV) ≈1,0
th`ı({ω×v(ψmaxLV)}/ψmax)≈1− ψmax−ω
ψmax−ψmin
Di.nh ngh˜ıa 3.6 Cho da.i sˆo´ gia tu.’X = (XXX, G, H,6), v l`a h`am di.nh lu.o ng ng˜u ngh˜ıa cu’a XXX φk : [0,1]→XXX go.i l`a h`am ngu.o c cu’a h`am v theo m´u.c k du.o c x´ac di.nh:
∀a∈[0,1], Φk(a) =xk v`a chı’ a∈I(xk), v´o.i xk∈XXXk
V´ı du 3.6. Cho da.i sˆo´ gia tu.’X = (XXX, G, H,6), d´oH+ ={ho.n, rˆa´t} v´o.i ho.n < rˆa´t
v`aH−={´ıt, kha’ n˘ang} v´o.i´ıt>kha’ n˘ang, G={nho’, l´o.n}. Gia’ su.’ choW = 0,6,f m(ho.n)
= 0,2,f m(rˆa´t) = 0,3,f m(´ıt) = 0,3, f m(kha’ n˘ang) = 0,2
Ta c´oP2 ={I(ho.n l´o.n),I(rˆa´t l´o.n), I(´ıt l´o.n), I(kha’ n˘ang l´o.n), I(ho.n nho’), I(rˆa´t nho’),
I(´ıt nho’),I(kha’ n˘ang nho’)}l`a phˆan hoa.ch cu’a[0,1] f m(nho’) =0,6, f m(l´o.n) =0,4, f m(rˆa´t l´o.n) =0,12, f m(kha’ n˘ang l´o.n) =0,08 Ta c´o|I(rˆa´t l´o.n)|=f m(rˆa´t l´o.n) =0,12,hayI(rˆa´t l´o.n) = [0,88, 1] Do d´o theo di.nh ngh˜ıa Φ2(0,9)= rˆa´t l´o.nv`ı0,9∈I(rˆa´t l´o.n)