Tifp chi Till hgc vk Didu khien bgc, T.28, S.l (2012), 88-102
MOT SO VAN DE VE PHU THUOC DA TR! TRONG CO Sd DUT LIEU Md CHUfA DUT LIEU N G O N N G Q
LE -Xl .AN \IXH, riiAN T H I S N THAMf
Ti-udng Dai h-QC Quy Nhan Email: lexninivinh(<^qnu.cdu.vn
Tom till. Biii liAo dua ra duih nghia phu thiiye da trj trong cd sd di? li^u md chila dii lieu ngon ngii dUa tren quan difim sii dung quan he tifOng ty duyc xay dijng nhd phan hoach tren mien trj ciia thuOc tlnli ling vdi mOt d^i s6 gia tu: thfrh hop. C ^ tinh ch&t ciia phu thuOc da trj mdi dinh nghia se dupc nghi&n ciiu v;i cu6i ciing m^t t^p c6c quy tSc sii> dien dii(?c chiing minh la diing din va day du tren m^t ldp c^c IU(?c dd quan h§ thda man mc>t s6 di^u kifn nhat dinh.
Abstract. In this paper, we present a definition of fuzzy multivalued dependencies in fuzzy databases with hngiiistic data based on similarity relation which is built by partitioning the domain of attribute values corresponding to an appropriate hedge algebra. Some properties of fuzzy multivalued depen- dencies are studied. Finally, the set of inference rules is shown to be sound and complete for a relation scheme class when it satisfies some specified conditions.
1. G i d l T H I E U
Mic'ii tri cua c^ thugc tinh trong cO sd dfl li?u (CSDL) quan h? kinh chen duOc gia thi^t chi bao gom c ^ gi^ tri ro. Tuy nhien, trong thi^c t^ dii lieu c6 t h i bao ham nhung thdng tin md, khdng chinh x ^ , thdng tin dtrdi dgng ngdn ngii. \'iei' xii ly c ^ thong tin n&.y vugt ra ngoai kha nang cua ly thuyet cd sd du" lieu kinh diln. Co nhieu c&ch tiep can dg giai quy^t vin de dat ra nhU: sir dung tap md [4|, dimg phftn phoi kha nang [12| hoac su: dung quan hg tuong ttr [2, 11]. Chi tiet hOn, trong [1], cae t&c gia da chuyin ddi ckc g\k tri md, g\k t n ro ve c ^ khoang so vk xay dung quan h§ gaji nhau ngii nghia (semantic proximity) ttt kich thudc cua c6c khoang sd. tir do dinh nghia phy thupc h&m va phu thugc da tri dua tren quan h6 nay. Trong [11] cdc tdc gia da bilu dien thong tin md b ^ g mgt tap c ^ gid tri ro va xdy dimg quan he tuong ttr tii khoang cdch giOa cac tap dugc bilu dien. Va phu thuoc da tri md tren cd sd diJ lieu ngon ngii ciing ditdc c ^ tdc gia dinh nghia dila tren quan he bang nhau ngudng a dugc xay dmig ttt mgt dg do khoang cdch gifla hai tap md trong [5].
Dai s6 gia tti (DSGT) la mpt trong nhOng each tiep can dk xay drmg quan he tUdng tu trgn mien tri chda gid tri ngdn ngii cua thupc tinh. TVong [9], cluing toi da de xu&t cdch bilu di§n nhieu dang dfi li?u khde nhau b ^ g tap cdc khoang trong [0,1] va xay dung quan h§ tudng tii dua vao cdc khoang md tren DSGT tUdng iing vdi mien tri ciia thugc tinh. Khi do cdc d^ng
MQT SO VAN DE VE PHU THUOC DA ' n t l TRONG CO sd OC Ll$ll U6 CHLfA DO LIEU NGON NG0 89 diili^u nhu s6, khoang, ngon ngfl cung nhir ejie d^jig ddc biOt nhir "missing", "inapplicable",
"at present unknown",...da dirge bilu dien mgt each thdng nhat va khdi iiicun bdng nhau md miic k dUdc dmh nghta lam cd so do xay di;ng cdc quan h? doi sdnh, chuyin cdc truy van mc) sang truy vdn rd.
V6i cdch ti6p can bdng DSGT, phu thugc ham md tren CSDL md chda dii lieu ngdn ngfl da dugc dinh nghia va nhi^u iinh chat quan trpng da dupe trinh bay trong [10]. Tuy nhien, phu thupc da tri md theo cdch tiep c ^ nay van chira dupe tiuan tam nghien cfm. Trong bai bao nay, tren cd sd khdi niem bKng nhau md mflc k chung loi de xuat djnh nghia phu thuoc da tri md; chiing minh mgt so tinh chat quan trpng ciia 116 vd euoi cimg mpt he cdc quy tdc suy dign lien quan den phu thupc ham md vii phy thuoc da tii md dupe chflng minh Id dung din va day dii tren ldp cdc hrpc dC> quan h? thoa man mgt so dieu kien nhat dmh.
Ngodi phan gidi thieu va. ket lu§n, bai bdo dugc td chflc nhir sau: Myc 2 ddnh cho viec trinh bdy mo hinh CSDL md chfla dfl lieu ngdn ngfl vd (luau he bang nhau mflc fc; Muc 3 tnnh bay nipt so ket qua qumi trgng ve phy thupc ham ind; Muc 4 gidi thieu dinh nghia phu thupc da tri md vd chihig minh tinh diing ddn ciia mgt so quy tdc suy dien; Muc' 5 danh cho viec chflng minh ti'nh di\y dii cua mgt tap cdc cjiiy tdc suy dien doi vdi ldp cac lirdc dd quan he thoa man mpt s6 dieu kien nhat duih Ndi dung chinh ciia bai bdo t^p trung d hai muc 4
2. MO H I N H CO s d D C LIEU M C C H I ? A UlJ LIEU N G O N NGLf VA Q U A N H E B A N G N H A U M L T C k
CSDL md chfla dfl lieu ngon ngfl Id mot tap DB bao gom {U,Ri,R2, ...,Rm;Const}, d day [/• = {Ai,A2. .,v4n} la khong gian cac thuOc tinh, mdi Hi la mgt lUdc d6 quan he, Const la tap cac rdng buQc dfl heu. Gpi mien tri cua mgt thudc tinh A nao do (trong so cdc AJ la 'SA- D6i vdi mot so thugc tinh, ngodi cdc gia tri thong thirdng IJA con chfla cac khdi niem md va nhiing khai niem nay la cdc phdn tfl ciia mot DSGT. Khi do S ^ co hai phdn: phdn thfl nhat bao gom cac gia tri thdng thudng dupc goi Id mien tri tham chieu, ki hieu bdi DA;
phan thfl hai bao gom cdc gid tri ngon ngfl cua DSGT tUdng flng vdi A, ki hifu bdi LDom{A) (6 day A ddng vai tro nhu mgt bien ngon ngfl cd kh5ng gian cd sd chinh Id DA) • ,Thu6c tinh nhu vdy co LDom(A) ^&vk A diidc goi la thuoc tinh ngon ngfl va ta gpi CSDL dang xet la CSDL md chfla dfl lieu ngdn ngfl hay ggi t^t la CSDL ngdn ngfl.
Vi du 2.1 Mpt quan he R cua CSDL ngon ngfl M a N V
01 02 .03 04 05
H o va Ten Nguyen Ngoc Vinh Le van Ninh Tran Van Himg L6 Thi Thanh Trinh Thi Tuy&
Tuai 49 45 27 tre r&t-tre
Li^dng rdt-cao 7.5 5.8 khd-cao
4.7
Phu-cap 4.6 4.2 [3.2, 3.8)
4.0 cao
Xet thupc tmh iuong trong vi du tren, gia sil thong tin khong day dii, chi bilt ludng cao iilSt cua nh&n vien 1& 7.5 trieu, thap nhat bKng 0 {co nguM khong co ludng) va khang biet llldc ludng chfnh xac c i a mgt ai nguSi chi biet la doi vdi cac nhan vien dang xet tU hp co
90 !.!•; XUAN VINH, THAN TI-H1I;N THANH
Ifldng Tji^cao. fcliA-eao, Chung ta lliav rKiig jal-eao, Ahd-cao la eae khdi nipm md. Chung eo th(' bicMi dien dir(,)e bang cdc la,|) mi') hoae d day Id eae gid tri ngdn ngfl cua DSGT c6 tdp
|)lian lir smh Id {eao. tb4|)} vd lapeaegia tfl Id {if/, gan, khd, rdt} chdng han. Khi dd mien tn Iham ehieu D/^KOMR = [0,7,5] vd /./>»//(/,irciiig) [rat-cao, khd-cao, ...}. Nhu v^y, LDom{A) ehi chfla eac' gui trj ugciu ngfl la phan lU eiia DSG'I' tiidng flng vdi thuge tinh A. Nhung gid tri ngdn ngfl kliong Id phan tfl eua DSCTF nay khdng tliugc' LDnrn{A). Cdc gid trj ciia thuOc tinh Mii N\" la eae gid In ic'., C'ae gid (i j ciia Hinge tinh liu vd Ten cung Id c'de gid trj ro.
Chung Id nhfnig gid trj ngdu ngfl uhiriig khdng phai Id cdc khdi niem md nen khong phai Id phan lir eua nS( I F, \ i vav, Ll)ovi.{Mh NV) = 0 va LDom{HQ vk Ten) = 0.
TVong bai bdo nay, ehung la sc""- xc'-l, CSDL ngdn ngfl md rpng chfla cdc dgjig dfl hgu khde nhit: gid trj kho.uig, l.ip liflu li{,ui cae gid Iri ro, cdc kieu dfl lic'u khong xdc dinh, thieu, khong biet (uudefine. inaj)plieable, missing, uukncnvn) dd ch'' cap trong |9].
Xel CSDL md ehfla dfl lic;ii ngon ngfl DB vdi khdug gian cfc thupc tinh Id U. Mdt bp ( tren (' Id lubt mill x^t : Ai x • • • x /!„ -»• X)^, x • • • x 2)/i„. Cioi ( va s Id hai bg tiiy y, khi dd t{A] vd s\A] Id hai gid trj thugc S)^.
Chung ta bi^t rdng phu thiiOc bam vd phy thugc da In trong CSDL quan he (kinh diln) ludn luon can kiem tra sir bdng nhau eiia dfl lieu tri'-ii mi^n tn cua tliuOc tinh. Tudng tu nhii vay. d d.iv chung ta se nhSc 1^,1 quan diem de ddnli gid sir bdng nhau gifla t\A] vd s{A] trong CSDL ngon ngfl.
Gia sfl flng vdi thuOc tinh A Id DSGT I iiven tinh. day du, tir dc:. AX_ = (X, G, C, H, #, E, <) (xem chi ti^t trong [8|). \'di k Id mot so nguyen dUdng (fc > 0), goi A't = {x G X : |x| = A:}, tlic la tdp tdt ca cdc gia tri ngdn ngfl trong X_ md bieu dien chi'nh tdc ciia chung co dp dai bdng fc vd ro rang Xk+i = {hx E X_: x €. Xk, h £ H}. Mc'ii hx G X^+i co mgt khoang tinh md 3k+\{hx) xdc dinh nhd hdm fm |8|. Tdp tat ca cdc khodng ^t+iCia;) Id mpt phan hogch tren [0,1). Lan can cua mgt gid t n ngon ngfl llioat nhin cd the lay la mgt khoang nay, tuy nhien "kich thudc" khoang md ciia mdt gid tri ngon ngfl y phy thugc vdo dg ddi cua no, c6 khi qud b6, c6 khi qud Idn khong ndm trpn trong iiic)! thdnh p h ^ cua phan h o ^ h . Vi vdy, khdi niem ldn can ngfl nghTa loi thieu ciia y plm hpp hon diidc dinh nghia bdi mgt tap cac khoang ro phu hpp tren mien tn. ky hi^u Id Omin,jt(y)-
Gom cum ede khoang Dfc+i(^3:) mgt cdch thieii hpp (chi ixh xem trong |9l) ta c6 mgt phan hoach vd dfla tren phan hoach ndy chung ta xae lap mOt quan he tiidng tfl Sk- Hai phan tii sau khi <biidn hda tfl mien tri vao [0,1] cd gid tn ndm chung trong mpt ldp tiidng dUdng do phan hogch ndy sinh ra thi chung diipc ggi Id bdng nhau im'fe k
Nhu vdy, xet sii bSng nhau gifla t[A] vd s[A] cd the dimg Dinh ngliia sau ddy.
D i n h n g h i a 2 . 1 . [9] Cho DSGT tuyen tinh ddy dii AX vd dO md fm. Gia sfl VA la mpt ham dinh liipng ngfl nghia tren AX vd; vdi m5i fc ma 1 < fc < fc", Sk la mOt quan h§ tuong tfl mflc fc trgn DA- Khi dd, vdi hai bO « vd s tuy y tren [/, hai gid tri t[A\ va s[A] tren miin tri S)^
dfldc gpi Id bang nhau mflc fc, ki higu Id t[A\ =f^^k s[A\ h o ^ t[A\ =,, s[A], neu nhti tdn tai mgt ldp tfldng dfldng Sk{u) ciaa 5^ sao cho Omin.k{t[A]) C Sk{u) va 0„,n,fc(s[>l]) C Sk{u).
Dua tren Dinh nghia ndy, Menh dk 4.5 [9] cho chung ta tigu chudn kilm tra sii b ^ g nhau mflc fc gifla hai gia tri ngon ngfl t[A] vk s[A] tign ldi hOn: t[A] =jt s[A] khi vd chi khi v{t[A]) e 5it(s[>l]) hoac v{s[A]) e Sk{t[A]).
MQT SO VAN DE VE PHV THUOC DA VIU TRONG CO SO DLf LIEU MC) CHLfA DO LIEU NGON NGt7 91 Lien quan deu sfl bdng nhau eiia hai gid tri ngon ngfl vdi hai mflc khac nhau ta co khdng dinh sau day.
Menh d l 2 . 1 . Ni'n t[A] =k s[A\ thi t[A] =^.> s[A] vdi moi k' < fc.
Chiifng minh. Xeu fc' = fc thi khdng djnh Id hi6n nlni'ti. chung ta s('> chflng minh cho trudng hop fc' < fc. Gia sfl t[A\ =k s[A\. khi dd theo Menh S 4.5 [9], v{t\A\) € Sk{s[A]). Hdn nfla, do fc' < k nen Sk{s[A]) C 5A..(.S[>1]). X'i vay r(/[.-l]) e 4 ' ( s [ ^ ] ) . tflc la t{A] -fc' -s[>il.B
Theo tinh chat eiia DSGT. moi khoang md mflc fc + 1 bao gid eung ndm trong mpt khoang md mflc fc tflc Id 2k+i{h.r) C H^ (,r). Do dd. phau hogch mflc fc + 1 mjn hdu phan hoach mflc k. Nf'ii fc cdng ldn (dp md cang nhd) thi "kich thirde" ldp tUdiig dudng ciia qiian he tUdng tfl Sk cang nhd dan den diC*u kien khi so sdnh bdng nhau gifla hai gid tri ngon ngfl cdng chat. Vi vay khi fc —» oc thi hai gid tri bdng nhau neu cluing bdng nhau t.uvct doi theo quan he bdng nhau thong thirdng
Doi veil cdc thupc tinh ro (khdng d("' cgp den tinh md, tinh khong day du, khong chdc chdn) nhu ma nhan v\cn. hp ten nhdn vien trong \'i du 2.1 vd cdc thudc tinh ma mien gia tri ciia no khong clifla gid tn ngon ngfl Id cdc khdi niem md thi mflc fc ciia thuoc tinh la oo. Khi so sanh, ta sfl dung quan he bkng nhau thong thirdng, khdng xdy dung DSGT.
Mflc fc trgn mien tri ciia moi thugc tinh co thg khac nhau, flng vdi thudc tinh A ta vigt Id kA.
Mot tap {kA\A G U} se dirpc ki higu la ^u hoac cho gon Id ^. Chdng han, trong Vi du 2.1 fca CO thi chon mot mflc ^ - (oo, co, 1,2,2). Cac s6 1, 2, 2 la ba mflc tiiy chgn tUdng flng v6i ba thuoc tinh cudi. Xeu chpn Id 1 thi mdi cum trong phan hoach cua mien tri se la hdp cua cac khoang md cua mot so gia tri ngdn ngfl dg dai 2, ngu chgn la 2 thi mdi cum trong phdn hoach cua mi^n tri se la hdp ciia cac khoang md cua mgt so gia tri ngdn ngfl dp ddi 3 (xem chi tiet trong |9]). Vdi X CU,ki hieu Rx chi cho sii thu hep ^tiiU xudng X (hay xem nhu la mgt phep chieu ^ xudng X).
Tfl Dinh nghia 2.1, chiing ta cd khdi niem bang nhau cua hai bo tren tap thugc tinh
xeu.
Dinh nghia 2.2. Cho ^ = {fc^|A e U}, fc^ > 0 vdi mgi ^ e (7 vd X C (7. Hai ba t va s dirdc gpi Id bdng nhau mflc ^ tren X ngu nhu t[A] =kA 4^] ^^i "^"Ji Ae X.
Cho % = {kA\A e X} vk ^'x = {fc^j-^ e X} \k hai mflc tuong tfl tren X, ta ndi Ax > A'x hodc Id ^'x < Ax ngu vd chi neu kA > fc^ vdi mpi AeX.Tit Menh dg 2.1, ta suy ra i - ^ 5 thi t =fc/ 5 vdi mpi fc' < fc.
Gia sfl X , y C t/ dang dupc xet vdi mflc Ax = {kA\A e X} va A'y = {k'g\B e Y} tUdng fing. Md rdng cua Ax vd A^, ki hieu Id AxVAiy, dimg dl chi mflc tuong tfl trong XUY theo nghia % V % = Ax-Y U Aly^x ^ -^Z- ^rong do 2 = X n F va K | - {fc^ V fc^^l^ e Z}, d day fc^ V fc^ = max(fcA, fc^)- Trtrdng hdp fc^ = fc^ vdi mgi ^ e ^ thi ta noi Ax va Aly tUdng thich nhau trgn Z vd khi do ta diing ki higu Ax U Ay thay cho Ax'^Ay-
Ttong bdi bao nay, ki higu .ffx A.^^. diing d l chi cho Ax-Y^^y-x^^Z' t'^o^S do Z = XnY va ^'^ = {fc^ A fc^l^ G Z}, 6 day fc^ A fc^ = min(fc>i, fc^)-
92 !,!•: XUAN VINH, THAN THIRN THANII
3 . P H U T H U O C H A M M 6 T R E N C d S6 DLf L I E U N G O N N G C Chiing la biel rdng tnmg sd c'de h)i,u plii,i tliiiOc dfl lien, phu tliubc hdm Id loai quan trpng
\;i difpe nghien euii iiliieu nhat, Plii.i tingle hdm ddng vai I id quan trpng trong \ lec thigt ke cd Ml dfl lieu. "Miflnj; smh vic'ti ed lic,)e lir< iigaiig iihau tlnrdn;; ed k(''t qua hgc lap gidng nhau" la mgt vi dn cua i)hi,i thupc bdm trong C'SDL ngdu ngfl. So ^anh "iigaiig nhau". "giong nhau" ro rdng kbcuig jihai la quan he; bang nhan llmiig tlnmiig iilui dfl Uen id Cluing Id cac khai nigm md vd phu lliune hdiii md Id rdiig biine lal yen dirg<' sir dijng Irong ngfl canh nav D i n h n g h i a 3 . 1 . [m] Cho DB Id ini)!. ed sc'J tlfl lieu iigc'in ngfl va /? Id mgt liipc dc> quan hg ciia DB vdi lap lliiipc' tinh U. Vdi X,Y CU va J? Id mgt mflc tiidng tfl tn'-ii U, ta ndi y phu tliiKie hdm mil va.i) .V mflc .li tren lii^c dA (luan lie /?, ki Inc-ii f = X -y^ Y, neu vdi moi quan h^ r ^ R:
( V ( . , s € : ) ( / | . V | = , i . , ( A - ) = ! . ( | y ] = , . , | r | )
Theo dd. lupe do c[ii,m he R se khong thda phu thiiOc hdm md / ngu tdn tai r G /? khong thoa / . Phu thiic'K bain md Iheo Dmh nglna 3.1 klidng pliai Id md rgng tam thiidng cua khai nigm phu thude hdm trong CSDL (juan be kinh clic-n Neu X -* Y Ik mpt phy thugc hdm thi ndi chung khdng ilic- ,siiy ra X —^^ Y Id phu lliii(")e ham md mflc A ndo dd; mpt quan hg r co cac bg ddi mcjt khac nhau tren X dong thdi cung khde nhau treu V Id vi du cho tnidng hpp ndy.
Xgifde lai, iic''u X —t^ Y la mgt phu thupc hdm md thi noi chung khong the suy ra X ^ V la phu thupc hdm. \'i du hai bo (a, 6), (a,6i) G X x K vdi 6 / 6i nhung b =^bi.
Gia sfl F Id mot tdp cdc phu tliiic')e hdm md tien R. Phu thugc hdm md / = A' —>^ Y dUdc ggi Id he qua logic cua F neu nhii vdi bat ky r e R.T thoa cdc phy thudc hdm md trong F thi r cung thoa / va ki hieu \k F \- f\ tdp cdc he qua logic suy ra tfl F se dUdc ki higu Id F', tflc F* = {f\F V- / } , Tuy nhien, chflng minh f ^ F' theo dinh ngliia la mgt bdi todn co dp phflc tap hdm mu theo s6 thupc tinh trong A i " M vav. mgt tap cdc ciiiy tdc suy dien dUdc dg nghi trong |10| d l giai (iiivOl bdi toan ndy mot cdch hieu qua hdn:
(Kl) Phdn xa: X ^^Y neu Y CX.
(K2) Md rong: X ^^Y ^ XZ -^^ YZ, vdi mgi Z QU vd Aid md.rqng cua A tren XYZ;
(K3) Gidm m-dc: X -*-^ Y ^ X ->j^' 1' neu A!^ = .IIA vd A\ < .^y- ,•
(K4) Bdc cdu: X ^^Y,Y -^n- Z =f- A ->jivj*' Z vdi Aly < Ay.
Mot phy thuOc ham md / dugc ggi la suy dien dflpc tfl F bdi eac quy tdc sin- dien (Kl)- (K4), ki higu F |= / , niu nhii ton tgi mdt chiy cdc phu thugc ham md / i , . .,f^ ^ f sao cho vdi moi 1 < i < m, hodc / , G F hodc /, suy dien dupc tfl cdc phy thugc ham md trong { / l i - " , / . - l } hdi (K1)-(K4). Tap t^t ca eae phy thudc ham md diipc suy dien tfl F bdi (Kl)- (K4) dupc kl higu Id F"*". l ^ p cde (|uy tdc suy dien (K1)-(K4) dd diidc chflng minh la dung ddn vd day du |1Q], tflc la F+ C F' vd ngUpc lai F " C F+.
D i n h ly 3 . 1 . [10] Gid sii F Id mot tg.p phy. thudc hdm. Khi dd F+ co nhiing tinh chdt sau:
(i}X^^YeF+^X^^Ae F+,\fA G Y.
(ii) {X ^^Y e F+,X ^^> Z e F+) ^ X -^^u^- YZeF+, vdi diiu kien .^x = ^ x - (iii) {X ^^Y ^ F+,V ^A. W e F+) ^ XV ^^vH' YW - F+, vdi diiu kien
^Ynvw - ^Ynvw-
MOT SO VAN DE VE PHIJ THUQC DA TRI TRONC CO s 6 D O I.IEl' MC) CHl'/A DO LIEU Nf;6N N G O 93 (iv) Dat G = {X ^^ A\X ^^Y & F vd A EY}. Khi do, G+ = F+.
Ttt cac quy tdc suy dieu (K1)-(K4) vd Dinh ly 3.1 ta suy ra mpt so tpiy t^\e khde. DC- tien cho viec trinh bdy phan sau eua bdi bdo, chiing ta se danh so cdc quy t ^ ' trong menh de sau la (Kil), (K12), (K13).
Menh d l 3 . 1 . Cac quy tdc suy diin (Kil), (K12), (Kl.'i) .sau day la diing ddn (Kil) Quy tdc hop: {X -^^ Y, X -s-j^- Z} \= X ^^yj^, YZ vdi Ax = A'^- (K12) Quy tdc gid bdc cau: {X -^^ V". \VY -^n' Z} ^ WX -^^^A- Z vdi A'y < Ay.
(K13) Quy tdc tdeh: Neu A -^^ Y n) ZCY thi X ^,,i Z.
ChiJng minh. (Kil) ehinh la khdng dinh (ii) eua Dinh ly 3.1, ta se chflng minh cdc quy t &
con lai.
(K12) Tfl gia thicM A ->^ 1' vd (K2) ta suy ra i l ' A -^^ WY vdi md rgng tuy cua A len W, chdng han lay Aw = .tt,,. Kc''l hpp vdi dieu kien A'y < Ay ta co A!y^ < Ayw Vi vay theo gia thiet WY -^^' Z va sU dung (K4) ta thu dUdc Vl^A -^RVA' Z.
(K13) Dg dang suy ra dirdc bang cdch sfl dung (Kl) vd (K4).
4. P H U T H U O C D A T R I M 6 T R E N CO S6 DLf LIEU N G O N N G U Trong CSDL kinh dien, phu thupc da tri diidc nghign cflu vdi muc dich trdnh mgt dang du thfla khi md hai tap thupc tinh dgc lap lai phu thugc ngfl nghia vdo mpt tap thugc tinh khac.
"Khi nhieu sinh vign cung thflc hien chung mdt dl tdi, neu mgt sinh vien duoc cho dilm 9 thi tdt ca cde sinh vien con lai cimg thflc hien dl tai do diu dUdc cho digm 9". Nhu vay, sinh vien vd dilm Id hai thudc tinh ddc lap phu thupc vao dl tdi. Vdn de cung xay ra trong CSDL ngon ngfl, vdi rang buOc mgm deo hdn, chdng han "Khi nhilu sinh vien ciing thUc hien chung mot de tdi, neu mdt sinh vign dupc dilm 9 thi mdi sinh vign cdn lai cimg thflc hien de tdi do deu CO mflc dilm tUdng dudng vdi dilm 9". Trgn quan dilm bang nhau mflc fc dd ndi trong Muc 2, chung ta se dfla ra dmh nghia phu thuoc da tri md trong CSDL ngdn ngfl va nghign cflu nhiing van de ddt ra.
Xet ludc do quan hg R cua mdt CSDL ngdn ngfl DB vdi tdp thupc tinh U = {Ai,..., A^}, trong dd n > 3. Dg cho gon ta se dimg ki higu XY chi cho A U F d6i vdi hai tdp con tuy y X,YCU.
Dinh nghia 4.1. Cho DB la cd sd dfl ligu ngon ngfl vd R Id mot lUdc dO quan he cua DB vdi tap thupc tinh U. Gik s^ X,Y C U, Z = U - XY vk A ]k mot mflc tUdng tii tren U. Ta ndi R thoa phy thuSc da tri md X -^->R Y niu nhii vdi mpi r e R, bit ky hai bd h>t2 G r c6 ti[X] =^ t2[X] thi trong r cung t i n tai bO ^3 vdi hlX] =^ ti[X], ts[Y] = s ti[Y], HZ]=^t2{Z].
Vi du 4.1 Xet quan he r nhir sau;
[.V. X U A N V 1 N 1 I , T R A N 'I
'II d c I
CSDI.l CSDI.I CSDI/J CSDI.L'
I ( T ) Dl
r.'il-lol S
( D )
Ten dc liii
l l l i r l l t i l ClIll <
cAc ph;\n lir i
( I ) va sinh vicii (.S) lfl cir tliiipc tliili ro, diciii (D) \h lliiiiic ti'nh md. Trong iciii (11 chiin 7.11 Inl, khli lol Din \h iiic l<liiU niCm inrt Ini'ii (hc'ii AiKfc bang na DSCI', Do d6 LDnm(D) / 0 va clifiii la Ihiioc thih iigoii iigO:. Gia sii inii'ii III lliaiii chini ciia D , U(n) = (I), l()|. Caning la so xiiv tlijiig D S t r r luven tinh .l.V = ( V. (;. (', H, <) (Ic bii'ii dii'n clio LD(nn(n) nhir san G = (li>l. xait], C = (ll, W. 1), H = II U / / + = {gin, it] U {khd, ull], Irong (16 if > nan vii rut > Wiii; c4c tham .MS m6 / m ( ( 6 t ) = /m(iAuj = U" = 0.5, /j((;iiii) = (i(iO = //(fchA) = /i(rat) = 0.25. Gia sir mi!c tirong li.r cila thilQC tinh dii-iii (hrrjc ch(?ll bang 1.
IVirdc ticn, chung ta tinh giA tri dinh liri^Ing cua cac gid trj ngon ngir vk chuygn d6i len mien tri tham chieu D(D) = |0,10| vdi hO so r = 10 bang ham dinli lirijng ngfl nghia |8]:
c, (1(A) = \W + 0.5 X /m((6t)] X 10 = |0.5 + 0.5 x 0.5| x 10 = 7.5
Vr(it-t6t) = \W + 0.5 X /m(«-io()l x 10 = (0.5 + (1.5 x 0.25 x 0.5] x 10 = 5.625 Dr(jan-t6t) = |H' + /m(«-((ji)+0.5x/m{..jan-t6t)] x 10 = [0.5 + 0.25 x 0.5 + 0.5 x 0.25 x 0.5] X 10 = 6.875
t),.(*;ha-tet) = »r(i6t) + 0.5 x fm{k\ik-toi) x 10 = 7.5 + [0.5 x 0.25 x 0.5] x 10 = 8.125 i.,(rat-t6t) = 100 - 0.5 x / m ( r a t - t 6 t ) x 10 = 100 - 0.5 x 0.25 x 0.5 x 10 = 9.375 Bay gici, chung la tinh mot so khoang tirong lu ticn |0,10| tao thanh b(ii cAc khoang md ciia cAc gia tri ngon ngfl co do dai 2, chfla cjic giA tri can xtH:
S^.ritU) = 3r(.9in-t6t) U 3r((tha-tSt) = (7.5 - 0.25 x 0.5 x 10,7.5 + 0.25 x 0.5 x 10] = (6.25,8.75]
Si,r(rat-t6t) = (10 - 0.25 x 0.5 x 10,10] = (8.75,10]
Si,r{it-tdt) = (0.5 X 10,0.5 x 10 + 0.25 x 0.5 x 10] = (5,6.25]
Cuoi cung, ngu chpn fc = 1 thi quari he tiroiig tfl 5i da phan hoadi nfla tren mi^n tri tham chigu cua thuoc linh didm thanh cAc khoang (5,6.25], (6.25,8.75], (8.75,10] chfla it-tdt, kha-tot, rat-tot tflcJng flng. So sAnh vdi gia In dinh Iflgng da tinh 6 Iron, ta suy ra rat-t6t —t 9, fcha-tot =(c 8, fcha-t6t ^k 0-
Miir vay, neu chpn mflc ^ = (oo, oo. 1) thi quan lip r da cho llioa phu thupc da tri md T
—>—*li S (va T —>—>ii D). Khi tliay digm cua sinh vien C bang 9 thi quan h^ thu dfldc khdng thoa T ->->ji S (do fcha-tot ^k 9). Rd rang r khong thoa T ->-» S theo dinh nghia phu thupc da tri trong CSDL quan hp kinh dign. Va vi du da minh hpa mot kiSu rang buoc md "Nhiing sinh vien cting thflc hien chung d^ tai thi diem ciia tat ca cA^ sinh vien do pliai tfldng dfldng nhau".
N h a n x e t 4.1 Do vai tro ciia fi va t2 trong Dinh nghia 4.1 nhu nhau va gia thi^t ti[X] =ii ^2].'^] nen ngu r thoa phu thupc da tri md nhfl tren thi trong r cung ton tai mpt bp t4 vdi ti[X\ =^ ti[X], ti\Y] =ji fa[y], ti[Z] = « tilZ].
M O T SO VAN D E vfi PHV THU13C DA TRI TRONG CO s d DO LIEU Md CHLfA DO LlfiU N G O N N G O 95 Chung ta ciing quy flde rkng ki hieu t\[X] =,1 t2[X] dong nghia vdi phat bigu ti b ^ g t2 mflc ^ tren X, vdi bat ky X C U. Tfldng tfl nhfl trfldng hpp phu thudc ham, bay gid chung ta se dfla ra mpt s6 quy tac suy dien chp phu thupc da tn md.
Menh dk 4.1. Tap cdc guy tdc suy diSn (K5)-(K10) sau ddy la dung ddn:
(K5) X -^—^R Y =^ A' ->-*ji' 1' vdi moi R' sao cho ^'^ = ^x vd ^'u_x - ^u-x- (K6) X -^-in r «• X ->->ji (£/ - XY).
(K7) Md rong X ^ - > , i Y, Z C W => .V\l' -»-»« YZ.
(KS) X ->->s Y, Y ->->«. Z => A' -|->K" Z-Y vdi R'x = Ax, Hy £ -Sy "" -S" = -SAi?'.
(K9) X ->ji y => A ->->« Y.
(KlO) Neu X - n , i 1" t>d Z C 5'. 11' n 1' = 0, Ax = S!x, A'u-x £ ^u-x thi W -t„f Z keo theo X -*ji' Z.
Chiing m i n h . (Ka) Gia sfl tdn tai mpt quan he r e R(U) thoa X -»->« Y nhflng khdng thoaX -^-^f^• Y vdi J?'x = .i?.\ vaiSy_Y i •'<' - f Khi do tdn tai ti,(2 £ r c d t i | X | =ji- tilX]
nhflng trong r khdng tdn tai mpt bp rs nao thda man r^lX] =si' ti[X], ri[Y] =^ t\^\, T-i\fJ - XY\ =,,( t2|C - XY\. Ta se chflng minh dieu nay vo ly.
That vav, vi .S'^ = Ax nen ti[X] =« t-i\X\. Do r thoa X ->->« Y ngn trong r tdn tai mgt bo'ta cd ts\X\ = « ti[X], ts\Y\ =j, ti[y], tz\V - XY\ =g tiW - XY] thuoc r.
Tit gia thilt Au-X > ^ ' y - x ^^ ^ ? " ' ' ''^ •^•'' ' * ^^^ ™ '^''''^l ^ ' l ' ' i W ' 'sl^l " " ' _*lM' t3[!7 - XY] =f. t2\V - XY], thda cac digu kign cua n lai thuOc r. Digu nay mau thuSn vdi gia thigt phan chflng. \'ay X —>—>>(' Y.
(K6) X - > ^ s y « X - + ^ s ((/ - x y ) :
Dat Z = t/ - x y . Xet bat ky quan hg r e fl va hai bp (1,(2 6 r cd ti[X] =« «2[X]. Vi X ->->j, y nen theo Nhan xet 4.1, tdn tai (3 £ r vdi
t3(X] =,i f2[X],(3[y] =« «2iy],(3[z] =« fiiz]
Vi t2[X] = s ti[X] nen t3[X] =ii ii[X] va vi vay (3 c6
t3[X] =Rti\X],ti[Z] =s(i[Z],(3[y] = t 2 [ y i
va (3 E r. Theo Dinh nghia 4.1, X - > ^ « Z. Vi vai trd cua y va Z nhfl nhau nen ta cung suy ra dflpc neu X ->->jj Z thi X -^->ji Y.
(K7) X -»-+), y , Z C W => X W - > ^ ; , y z :
Gia sfl tdn tai quan he r 6 B va hai bd (i,«2 £ r cd ti[XW] =SL t^lXW] nhflng b i t ky ba r3 thda man
r3[X»'] = , (i[X»'],r3[yZ] = , (i[yZ],r3[£/ - XYWZ\ =« (2[£/ - XYWZ]
lgi khong thudc r. Chung ta se tim mAu thuSn ttt gia thigt nAy
ThAt vay, i i [ X » ' ] =« f2[XW] keo theo ti[X] = « (2IX]. Do X ^ - > i , Y ngn tdn tsi <3 e r
CO
tsl^] =« n[x],i3[y] =ii ti[i'],t3[c - ^i'l =ii «2P - ^^1
96 1.1; XUAN VINII.TRAN THIiiN T H A N H Tfl hai ilaiii; Ihiic (lausiiv la t3[Al-nM'] = „ t,lXYnW] Tn lid cd
'ii|f/ - A I'I = „ bill/ - x y ] .* i:,\w - XY] =„ I2IW - x y ]
\-ii
d l A l l I - , , (.[All--] = > / , | [ M / ] = f l ( 2 [ H / ] = i - « , [ H ' - X y ] = « ( 2 [ » ' - X y ] Dodo. (,i]li" A l ] i | / | | i l ' A>'], Dien nay ki'l hdp v d i / ( ( A y n l V ] = j t ( i [ X y n H ' ] la ,siiv ra r , | i r ] - . , r,]irj \'a vi vav dilAU'] = „ ti[XU'] vA /,[Z| -n t,\Z\ do Z C W. Ket hpp vdi l,,\y] = « ( i l i " | t a c d (i[l'Z] =,ft,\YZ].
Biiv gid tfl t;t[U - v y ] =,, r^l" x y ] (1(' dang snv rn dflpc t:,{U - XYWZ] =« (2[(/ - .Vi'U / ] Nlifl vay. '3 Ilioa cac ilicii kn-u ciia r;, vk /,{ e r. Dion n;tv man thuan vdi gia thiet phan chflng. \'.i>' -VU' -^-i,E YZ.
(K8) A -»->fl 1'. )• -»->„. Z => X -t-»j,./ Z - y vdi dii'-ii kifn Rx = Si\. Ay > -Syva It" .11 A .It':
Gia sir li'iii lai /• G /? va hai \>i'iti,f2&r (d ti{X] =n" (2]X] nhflng bat k.v bd r3 thda man
'•i[A] =«" ii[x],r3[z- y] =«.. (i[z - y],r3[u - x ( z - y)] =«" i2[u - x ( z - y)]
lai khong tliuur r.
Xl A'^ = Ax nen t\[X] —^ i2[-^]. theo gia thigt X —»—J-^ 1' vd Nhdn xet 4 1. ton tgi mpt bp (3 6 r cd
t3[X] =s (•lX],t3[yl =« t2lY],t3{U ^ XY] = « ( , [ ( / ^ XY]
Ttt J?'^ = Ax, Ay < AY va. A" = AAA' tA sny ra A'y = A'y va .15"^, < .itj _, nen theo (K-j) ta cd y —>—*j," Z. va vi vay ton tai (4 £ r cd
(4[y] = « " i2[y],(4[z] =„" i3|z],«4[t^ - YZ] =ff. (2|c^ - YZ]
Ta s(^' cluing minh mdt so khang djnh sau dAy:
' t4{X] = s " ti[X]: That vAy, ttt(4[Z] =,,» i3[Z],t3[X] =jiri|.A] va A.v = A'x, ta suy ra
(4[XnZ]=«» f,]xnz]
Hdn nfla, lir (4]!/ - YZ] =„» t2lU - YZ], l,[Y\ =«.. (2[y] va (i|X] =ji» (2[X], ta lai cd f4[X - Z] = , , . ( , [ X - Z ]
Vivay, (4[X]=;,»(i[X].
+ f4|Z - Y] = s " (i[Z - y ] : That vay, t^lZ] =R„ t^lZ] =s. t4[Z - Y] =„» f3[Z - y ) . Hdn nfla, ttt
*3[X] = „ (i[X], t3[U - XY] = ^ tliu - XY] va A" = A A A' ta suy ra tslZ ~ Y] =«» fi[Z - Y]. \ a v (4[Z - Y] =«.. f,[Z - Y].
4 tiiu - X ( Z - Y)] =;i» (2[C/ - X ( Z - y ) ] : M t4[Z] =;,„ (3[Z] theo gia thilt vA t3[y] =s t2[Y] => (3lyl =«" t2[y] nen
«4[y n z] =j5,, (2[y n z]
MOT SO VAN DK VE PHU THUOC DA TRI TRONC C(i S(i 1)1? LUU Mi) CHITA Dif LIKI) NCON NCO 9 7 DS y rSiig ((/ - X ( Z - y ) ) h Z = ( y n Z) - A. (lio n(--n
<4[((/-A(Z - y ) ) n Z ] = f l „ r 2 [ ( ( / - A ( / - i ' ) ) n z ]
Hon nfla, ta ciing cd bao liiuii liip hpp {U - .\{Z - )')) Z C (// l ' . V ) U i ' va llicogia tliii'l, t,[Y\=g-t2\y].lilU- I'Zj = „ » ( , [ ( / - ! / I nen
( 4 [ ( t / - A ( / i ) ) - Z ] = „ . . / 2 [ ( ( / - .V(Z 1 " ) ) - Z ] Viiy tiP - X(Z - 1)1 =,.. (2[!/ - A ( Z - Y)\
Xliir vay. /] tlida man cac dicu kien cua rj vii ri £ r. Dicu ii.'iv inan Ihuan vdi gia llii(''l phan chihig. \.iv .A ->->.ii" Z - Y
(K9) .V ->,i Y => .V ->-»,,[ 1" Gia sii; liin lai r € /f((') lliiia .V -»u V nhimg khdng thoa .V -)->,i 1'. nghia la u'lii lai (i, /j £ r (o ri [.V| =,i /jj.V | nhimg lial k\ bo 7-,t thda man nlX] =,! /ilA|, rajy] =« (i[l-]. r , | ( ' - X l ' l =,i (2|f/ - A l l 1,11 klKiiig IhiKic r.
Xet bp f2. ta cd (jIAj =a (i[X]. (2li'] =.i ' i | y ] vi .A ->« V vii hifn nhien (2IC - XY] =a t2]U - XY]. Xliir vay t, thoa e.ic didu kien cua Ti vii /j £ r Di(\l niu man thuan vdi gia thiet phan chflng. \'.\\ -A —>—>n Y
(KlO) XiSi .A -(->« 1' va Z QY \V D 1' = 8. .S.v = A^y, .it, _ \ > .lij y Ihi II' ->«. Z keo theo A' —>,i' Z:
Gia sfl r £ R{U) thda A ->->,; V va II' ->ff Z. a day Z C Y v'n W nY = Vl nhflng r kliong thda A ->j^' Z. nghia la tdn tai ti,t> G r cd ri[X] —a' ^2[X] nhflng (i[Z] ^a' t2[Z].
Do X -»->^ y vit A.v = A'Y nen tdn tai fa e r cd
f4[x] =s (i[x],(4[y] =;, (2[y].(4l£/ - XY] =g ii[c; - x y ] Kit hpp cAc khAng dinh niiv vdi gia thiet IT n 1' = 0 , ta suy ra f i[ir] =15 / | [ i r ] . Ttt f4[yl = s (2|y] va Z C 1'. ta cd t4[Z] =« t2lZ]. Ki'l hpp vdi (,[Z1 ^fe tilZ] (gia thiet phan chflng) va .lt(--.v > AJ-_ y. ta suy ra t^lZ] ^^ fi[Z].
Baygid, ttti4[H^] =fi tilW] vAcAc dien kif'n Ax = A^,.lt(,r-,v > .'iJ'.. y tasiiy ra f4[ir] =ff tilW]. Xhir vay. (4IIV] =«. ( i | i r ] v!Lt,lZ] # « ' ' l | Z ] . Didu niii' man thuan vdi IT -+/;. Z. MiMili dl da dfldc chflng iiiiiili-B
Nhfl vAy, tinh dung dan ciia tap cSc (|iiv t;l( (Kl)'(KlO) da dflpc chflng minh va chi sir dung cac quy tac nay chung ta cd thd suv ra mdt sd quy tac tien dung hdn.
Menh dg 4.2. Ciic g«!/ tdc suy diin (KI4), (K15), (Kltj), (KIT) sau ddy la ili'mg ddn:
(KH) Quy tdc hap {X - » - t g Y,X -^-tff Z ) ]= X - • - • « " YZ vdi A!' = Ah A' neu Xx = Ax vd (Ay < Ay hoac Az < A'2).
(KIS) Quy tdc gia hde cdu {X -•->„ Y,WY -»->«. Z} |= IT.V -»->«» (Z - ID") vdi X' = AA A ' neu A^^y = Axw vd A'y < Ay.
(K16) Quy tdc phdn rd
a) {X - n , i y , X ->->«. Z} 1= X -»->«» (Z - y ) i,(?i A " = A A A' neii A'x = Ax vd
•Sy < A y .
98 Li'^ XUAN VINH, TRAN TrnioN i IIANH
h) I \ ->-»„ 1". V ->-»„, Z} 1= V >->«" ( y - Z) vdi A" = A A A' neu A'x = Ax »d 11/ < Ji'/
cj (.V >->i,y. A >->,„z) I .V ->->,„ y n z
(Kn) Q«ii Ifir pha trim, i/iji. tac crm (X - i - ) , t y, X y -»;, Z} |= X ->ji Z - y Chi'fng iiiiiili.
(Kil) Ttt ,V -•->.„ )• ll IK7I la cd
.V -+^fl A y (1) l.ai Mrdi.ing (K71 voi gi.i lliii'l .V -i-^ji' Z, I.I suy ra X y ->->ji' YZ va d o d o lh(» (KG) taco
A ) - - » - • ; , . / ' A l Z (2) X(''ii .li'y = A.v v,'i A'y < .11) Ihi A'^y < )i\) \ 1 vAy, Ap dpng IKN) cho (1) va (2), ta cd
A ^ - ) , i " t / Al Z, vdi A" = A A A ' (3) Sir dpng (KO) cho (3) ta dirpc
X -»->«" YZ ~X (4) Lai sfl dung (K7) cho (4) vdi bp phan md rdng dirpc chpn la y Z n X ta thu dfldc X ->—ij^" YZ.
Trfldng hdp A'x = Ax va Az < A'z vifn chflng minh lioiin loiiii tirong tif.
(K15) Sir dung (K7) cho gia thiet .A ->-i,i Y thu dirpc
WA-»-»,! u y (5) Tir dieu kien ciia quv t ^ (Ki5), ta suy ra A'^yy < A u i Do vay ki'M hdp (o) va gia thilt
i r r ->-+«' Z, sfl dung (K8), ta dflpc II.V -»->«» (Z - H i ) vili A" = AA A'.
(K16) (a) Sir dung (K7) lan Ifldt cho cac gia thiet .A —^-^^ y . A —^—^R' Z ta co
A - > - > „ A y (6) vA
A ) - - • - » ; , , y z (7) Ttt cac dicu kien A'x = Ax va A'y < Ay, suy ra A'yj < Jt.vi-- \"i vav, sfl dung (K8) doi vdi
(6) va (7) ta cd
A -•-»;,» Z -XY (8) vdi A" = AAA'. Bay gid, md rgng (8) vdi (X - y ) n Z b^ig (K7) ta thu duoc X -»-»«» Z-Y.
(b) Dfldc suy ra trflc tilp tfl (a) nhd tinh dpi xflng ciia gia thilt.
(c) Tflpng tfl nhfl phin chflng minh cho (a) trong dd lay A' = A ta da kit luan dflpc
X ^ - » j , X ( Z - XY) (9) Md rpng X ->-»ji Z vdi Z - XY bkng (K7) ta cd
X ( Z - XY) ^-,R Z (10)
MOT S6 VAN DE VE PHU THUQC DA TRI TRONG CO Sfl DO LIEU M O C H O A D O LIEU NGON N G O 99 Sfl dung (K8) doi vdi (9) va (10) thu dfldc
X ^ - 4 « ( y n Z ) - X (11) Me rpng (11) vdi X n y n Z bang (K7) ta dugc X ->->„ y n Z.
(K17) Tfl X ^ - + H y va (K7), (K8) ta suy ra A ->-+g U - XY. Kit hop vdi gia thilt XY ->« Z va sfl dung (KlO) ta cd X ->« Z - XY. Hdn ntta theo (Kl), X ^ « ((X n Z) - y ) . VI vAy X ->j, Z - y theo Dinh ly 3.1(iii) vdi nhan xdt la (Z - XY) U ((X n Z) - y ) = Z - y .
Sau day chung ta se cliihig minh tinh d^y dii cua tap quy tac suy didn (Kl)-(KIO).
5, T I N H D A Y D U C U A TAP CAC QUY TAG S U Y D l I l N Trfldc tien IA mpt s6 dinh nghia hgn quan den vii^c chflng minh tinh day dti cua mot tap quy ti£ suy diln.
Dinh nghia 5,1. Cho F va G IA tap cAc phu thudc hAm md va phu thudc da tri md tfldng flng tren U. Bao ddng cua F U G , ki hieu IA [F,G)^, la tap cac phu thupc ham md va phu thudc da tri md suy didn dfldc ttt F U G nhd cac quy t ^ suy diln [Kl) - (KIQ).
Dinh nghia 5.2. Cho F va G la tap cac phu thudc ham md va phu thudc da tri md tfldng flng tren U. Vdi mdi X C Lf va A la mdt mflc tfldng tfl tren U, bao ddng X ^ cua X (flng vdi F vA G) la tap hdp {A e C/|X ->n >1 £ (F, G)+}.
Menh d l 5,1, X -^^ Y thuoc [F, G)+ khi vd chi khi Y C Xj[, d day X^ td bao dong miic A ciia X icng vdi F vd G.
Chuing minh. Dat Y — Ai...An vdi Ai,...,An la cac thudc tinh vA gia sfl ring X ^nY thuOc (F, G)+. Theo (K13), ta suy ra X ->s A thudc (F, G)+ vdi mdi i = 1,..., n. Va vi vay theo Dinh nghia 5.2 Ai £ X ^ ngn Y C Xp Ngfloc lai, gia sfl y C X j , tflc la A, 6 X j va X-»j,Aj thudc {F,G)+ vdi moi i = ! , . . . , « . Theo (Kil), ta suy ra X - > s y thudc (F,G)+.
Menh dg 6.2. Cho X C U, khi do tap U - X co thi phan hoach thdnh cac khoi Wi,...,Wm am cho vdi moi Z QU -X, X -»-»ji Z khi va chi khi Z Id hap eua mot so cac khdi W,.
Chihig minh. Diu tign xem toan bd (7 - X la mdt khli Wi. Gia sfl din mot thdi dilm nAo 46 (7 - X da duoc phan hoach thinh cAc khdi Wi,..., W„ (n > 1) va X ->-»n W, vdi mpi i = 1, ...,n. Niu X ->->ji Z va Z khdng la hop c i a cAc W, thi ta phan hoach mdi Wi thanh W.nZ va W, - Z. Theo (K16), ta cd X - > ^ j i M', n Z va X ->->s W, - Z. Vi tap thudc^tlnh V hflu han ngn cudi cung ta phan hoaoh dupc Lf - X sao cho vdi mgi X -^-^^ Z, Z deu la kpp cua mdt sd cac Wi. Ngflpc lai, niu Z la hdp cac Wi thi ttt (K14) ta suy ra X ->->ji Z.
Bay gid chflng ta se ph&t bifa va chflng minh tinh d&y du cua tap cac quy tac suy diln (Kl)-(KIO) del vdi mdt ldp cAc Iflpc dp quan he thda mdt s6 dilu Men nhat dinh.
Dinh ly 5.1. Vdi mdt mile A cho trrldc, tap eac quy tdc suy diln (Kl)-(KIO) la day du dSt vdi cdc luac do quan h€ thoa mdn dieu kiin: mien tri eua mdi thudc tinh Ai e U deu tdn tai
"i6t cap ai,bi sao cho a^ ^^^ hi.
100 l,P; \ | i \ N V I N r i , 'i'llAN T i l l K N ' I ' l l A N l l
111
('),),. fl ( " , ) , . fj_
( ' ' , ) . t f , (''.)(6f.
Il'i (",)„f, ( ' ' i ) , ( f ; ('.,)„ f, ('',),( 1.
. ''•"' i
(",),.-f,. ' (",),-'-f,.. ' 1 1(1,1.-Sf,,, (''•),ff,„
Chflng m i n h . \),- ilirtiiR iiiiiih Hull (lily (hi cua (Kl)-(KIO) di'ii vdi ldp ciic litdc dd quan he Ihiia man dicu kien iicil IK'II, la sr diinig iiiinli liliit; neu /•' vii G tfldng I'tllg lii tap php thugc li.iiii md vii phu Ihudl da In md (gi.ii (liiiiig hi jilii.i IIUKIC md) cho Inrdciua nipt Ifldc dd quan hi' iKMi l.ip lliiioc linh (' vii (/ h'l iiipl iiliii lliiioc md khoiig suv dien dflpc tfl F vA G bdi tjp ciiciiHv l.lc ll\l 1 IMII), lif( 1A(/^ (/•'.(.')' , III! Ion lai iiiol (|iian IK-r I hoa (fu |)hii IIIIUK md Irong {I--.G)' iilnnig khdng ll (/.
riial Viiv. giii sir d 1,1 php IhiKic md (o mflc A v.i vi' I nil l.i .V I lico .Mf'iih de ."> 2. ddi vdi .V t a c o lap V| 1„ la liKil phiin liiiaili tivii (f ,V I irdiiu fln;; vdi F vli G DAt X j = {4 £ ( |A -+„ ,1 £ (/••.(.')' 1 Khi dd X >,i A,; £ (F.f.') ' vA Iheo (K'l). A ->-*^ X* £ {F,G)*.
llici. (KO), la .suv 1.1 V > >,((' ,V,; £ (/••, f.'l. vA vi vav theo Mi'iih (!('•'. 2 thi ( / - X ^ la hpp (ua mdl so Ciic II, vdi ir, 6 |1 I. , V„). hi. IA ff - A,; ^ U " i " ' . Ha) gid. chung la «•
xiiv dflng ipiaii he r gom 2"' bo iiliir MIII X,|
(",)„ /., (".),.(„
(",),. i„
1 " , 1 , . I.,
Jorijf-jj,
\'i du vdi 111 = 3 va mdi tap -V,|. H i . W',. U'l chi diir.i 1 thudc tinh, r se gom S bp:
< a,a.a.(J > , < a.b,a,a > , < a,ii.h.a > , < a,a,a,b > , < a,b.b,a ^ , ^ a,b.a,b >. <
a.a.b.b >.< a,b,b,b >.
Trudc tien eliiuig la (liifiit; minh r thda F vA G. Gia sfl Y -»jj Z lii nipt plm thupc hAm md mflc A thudc F Dat II' = U„ , H'. vdi 7 = {i|y n II', / 0}. Dd llii'iv )' C . A t i r va vi vAy X t i r -»j, Z £ {F.G)-^ theo (Kl) vA (K4).
Theo Mi;iih d('- i 2. ta lai cd A ->->« 11' £ [F. G)+. Hdn iifta. .A ->->j, .A,t £ (F. G)+ nen sfldung(Kll) I a suy ra.A ->->,( .V,1 U £ {F.C)^ Theo (K171. .V ^ g Z - . V t i r E {F,G)*.
Theo dmh nghia ciia X j ta cd Z - A , ; i r C X,[ . Iflc IA Z - X,: II = 0 va vi vay Z C X j l l ' Gi.i sfl ti.fo £ r va ti[y] „ (2[y]- Theo cii.li xiiy dimg r. la cd (i]irl =x t,lW] vii ' i W l =Ji <2[XjJ], hii lii fil-V^jU'l =,i ( i l A , ! i r | , \'i v,iv (i[Z] =ii t2[Z], lire IA r thda Y^nZ.
B,i\ gid, gia sfl y ->->), Z la mgt phy thugc da In nid mflc A. tflc IA 1' - t - ^ j ^ Z eG. Vi 1' C X j l r va0 C .Y,Jir iicnsfldi,iiig{K7), ta dflpc X,t IT ^-.-..j Z £ [F.G]^. Hon nfla, theo chflng minh tren X -^^„ .A,; IT £ (F,G)^. dn viiy theo (KS). A -t^g Z - X J l i ' £ (F,G)-'' vA theo Mf-nh dd 3-.i.2 l.i suy ra Z - -V,tir = (J^^^^ ir, la hpp cua iiidt sd cac Wt vdi 7i C { l , . . . , m ) .
Vdi hai bd tiiy v (1,(2 £ r cd i i [ y | =n t^lY], tiic t,[y] = t2[y]. Theo cAdi xay dflng r, tdn tai mot bd t3 cd (,[Z - X+W] = d [Z - X+W] va t-ip - (Z - X+II') = t2lU - (Z - X+W)].
Biiy gid ta se chiing minh (3 IA bg thda cAc digu kien trong Dinh nghia 4.1. That vav.
do tslU - (Z - X + W ) = t2[lf -{Z - X+W)] va (2[y] = t i l y ] ngn t3]y] = tilY]; Tir '3[y] = ' l l ^ l ta suy ra t3(W] = fi|ir] va hai bd tiiy y trong r deu bang nhau trgn Xt ngn
MOT SO VAN DE VK PHU THUOC DA TRI TUONO CO Sfl DlT LIEU Mfl CHLTA DO LIEU NGON NGlJ 101 t3lX^W] = (i|X+H/] V i v A y t 3 [ Z n X + i r ] = / , [ Z n X j H ' ] Theo each chon (3, (3[Z-X+H'] = 1,[Z - X+W]. Do dd (3[Z] = (,[Z]. Cdn liii, (3]!/ - YZ] = fj]!/ - YZ] dfldc siiv ra ttt tiP - (Z - X+W)] = t2lU -{Z- X + W)] vdi nhiiu x(''l t/ - y Z C (/ - (Z - X^ H'). Xliir vay t3 thda cAc dieu kien troug Dinh nghia 4.1, tflc IA r Ilioa ) ' ->-iji Z
Biiy gid. chung la chimg minh r,*ing r khdng thda </ Thai viiy. xi'-l trirdng hdp d= X ^^Y.
Vid^ {F,G)+ ngn 1' ^ X,[ Theo each xiiy dl.mg r, li'm lai hai bd 11,(2 £ r iiia(,(y] 7^ (2[y];
trong khi dd f,[X] ^ f..,|.V] Ta sui' ra fi|A] =j, (.^[X] nhimg t,lY] / , fjly), tflc IA r khdng thoa d.
TYirdnghdpd= A - > - t « 1' i (7'. G)+. Dl chflng minh phau chflng gia sfl r thda d. Do cSch xay dflng r nen 1' = . V u l T ' . od,i\ A ' C X,'- vii i r ' = U,e.; ll''i-vdi 7 C {l,...,m}. Theo Menh dl 5.1, .V ->,t -A' £ (F, G)+ vh theo (K9) ta cd A ->->^ X' e (F, G)+. Tfl W = (J.^j W„
vdi J C {1, ...,m} v,i Menh de 5,3 la suy ra A ^ ^ j i r ' £ {F,G)+. Stt dung (K14) cho hai kfl luan tren ta cd d = .A - t - t j .\" u i r ' £ {F.G)+. tflc la d = X -t^n Y £ {F,G)+, man thuan vdi gia tiiict Wiv r khdng thda d. Dinh ly da dirdc chflng minli •
6. K E T L U A N
Trong bai bAo nay, cAc phy thudc cd bAn tren md hinh CSDL md chfla dfl heu ngdn ngfl da dfldc nghien cflu va dac biet la phu thudc da tri md vdi nhitng tinh chat cua no. Mdt tap cac quy t ^ suy dien dflpc chiing minh la dung dan vA day du tren ldp cac Iflpc do quan he thda man dilu kien nhat dinh. Dilu dang ndi la ttt each tigp can xem miln tri cua thudc tinh nhu mpt DSGT, tieu chuan bAng nhau mflc k la cd sd dl xac dmh sfl thda man cac loai phu thudc nhflng rat linh hoat thuan tien cho viec cAi dat vi cac tham so khac dfldc tinh toAn bdi ham dinh Iflpng ngtt nghia phu thudc trflc tilp vAo fc. CAc dang chuAn md lign quan din cAc phu thudc md trong bai bag se dflpc quan tam nghien cflu tiep theo.
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Nhdn bdi ngdy 15 - 11 • 2011
