I. H tin AmstrongBi Tp 1: Cho lc quan h R ( A, B, C, D, E, F, G ) v tp ph thuc hm F xc nh trn R:F ={ AB, D F, BF E, EF G, A C, BC D }Chng minh: AF G c suy dn logic t F da vo h tin Armstrong.Bi Tp 2: Cho lc quan h R ( A, B, E, I, G, H ) v tp ph thuc hm F xc nh trn R:F = { AB E, AG I, E G, GI H } Chng minh: AB GH c suy dn logic t F da vo h tin Armstrong.Bi Tp 3: Cho lc quan h R (A, B, C, D, E, G, H} v tp ph thuc hm F xc nh trn R:F={ AB C, B D, DC GH, HC E }Chng minh: BC G v AB E c suy dn logic t F da vo h tin Armstrong.Bi Tp 4:Cho lc quan h: Q(ABCDEGH) vi F= { AB C, B D, CD E, CE GH, G A }Chng minh: AB E v AB G c suy dn logic t F da vo h tin Armstrong.Bi Tp 5:Cho ph thuc hm F= { A B, BC D, AB E, CE G}. Dng lut suy din Armstrong chng minh: AC DG, AC E thuc F.Bi Tp 6: Cho G={ AB C, B DE, CD EK, CE GH, G AC}.Chng minh: AB EG bng lut tin Armstrong.Bi Tp 7: Cho lc quan h R (A,B,C,D,E,G,H,I,J) v tp cc ph thuc hm:F = {AB E, AG J, BE I, E G, GI H }. Tm chui suy din AB GH bng h tin Armstrong.Bi Tp 8:Cho R = { A,B,C,D,E,G,H,I} F= { A B, BH I, B D , D BE}. Chng minh: A E.Bi Tp 9:Cho lc quan h p= (U,F) vi U= ABCDEGH, F= { B AC, HD E, ACBE, E H, A D, G E}. Kim tra tnh ng n ca cc suy din ca h tin Armstrong: F|= CG EH.Bi Tp 10:Cho lc quan h LQH) P= (U,F), trong U= ABCDE, F= { A B, B E, D CE). Chng minh: AD BE bng lut suy dn Armstrong.Bi Tp 11:Cho R = { A, B, C, D, E, G } v F= { AB C , D EG , BE C , BC D , CG BD, CE AG}. Chng minh: AB CG da vo tin Armstrong.Bi Tp 12:Cho F={AB, CD} vi C B, hy chng minh AD suy dn c t F.Bi Tp 13:Cho lc quan h R(ABCD) v F={AB, BCD} hy cho bit cc ph thuc hm no di y c th suy dn c t F:a. ACD b. BD c. ADB Bi Tp 14: F={XYW, YZ, WZP, WPQR } Chng minh rng XYP suy dn c t F.Bi Tp 15:Cho lc quan h (=(U, F) vi U=ABCDEGHIJ v tp ph thc hmF={AB E, AGJ, BEI, EG, GI H} f=ABGH, Chng minh rng f suy dn c t F Bi Tp 16:Cho lc quan h = (U,F) vi U = ABCDEGH v F = { AE BEG , CEH BD , DG BCD, ABC DE} v mt ph thuc hm f = ACE DEG. Hy ch ra rng f c th dn c t tp F theo cc lut ca h tin Armstrong. Bi Tp 17:Cho lc quan h = (U,F) vi U = ABCDEGH v F = { AE BEG , CEH BD , DG BCD, ABC DE} v mt ph thuc hm f = ACE DEG. Hy ch ra rng f dn c t tp F bng vic ng dng cc lut ca h tin Armstrong.

II. Bao ng ph thuc hmBi tp 1: Cho lc quan h R = (U, F)U= {A,B,C,D,E,G,H}F= {ABC, DEG, ACDB, CA, BEC, CEAG, BCD, CGBD, GH}a) Tnh (D)+p n : (D)+ = DEGHb) Tnh (DE)+p n : (DE)+ = DEGHc) Tnh (BE)+p n : (BE)+ = ABCDEGHd) Tnh (CG)+p n : (CG)+ = ABCDEGHBi tp 2: Cho lc quan h R = (U, F)U = {A,B,C,D,E,G}F = {CG, BG CD, AEG BC, CG AE, B CG }a) Tnh C+p n : (C)+ = ABCDEGb) Tnh (B)+p n : (B)+ = ABCDEGc) Tnh (AEG)+p n : (AEG)+ = ABCDEGBi Tp 3: Cho R(A, B, C, D, E, G, H) v F = {AD, AB DE, CE G, E H}.Tnh (AB)+p n : (AB)+=ABDEHBi Tp 4 :Cho lc quan h = (U,F) vi:U = ABCDEGHF={ BC ADE, AC BDG, BE ABC, CD BDH, BCH ACG}a) (BD)+p n : (BD)+={BD}b) (ABE)+p n : (ABE)+={ABECDEGH}c) (CDG)+p n : (CDG)+={CDGBHAE}

Bi Tp 5 :Cho =(U,F); U=ABCDEGHF={ ABBCD, EBGH, ACD BG, DAEH}Hy tnh X+ trong cc trng hpa) (AC)+p n : (AC)+={AC}b) (CD)+p n : (CD)+={CDAEHBG}c) (ABG)+p n : (ABG)+={ABGCDEH}

III. Tm ph ti thiuBi Tp 1:Cho tp quan h R = (A,B,C,D,E,G) v F = { AC -> B, B -> ACD, ABC -> D, ACE -> BC, CD -> AE }.Tm ph ti thiu ca R.p n : {ACB,BC,BD,CDA,CDE}

Bi Tp 2:Cho R(A,B,C,D,E,I) v F = {AC, AB C, C DI, CD I, EC AB, EI C}Tm ph ti thiu ca R.p n : {AC,CI,CD,ECA,ECB,EIC}

Bi Tp 3:Cho R(A,B,C,D,E,G,H) v F= {ACB,CB,ABDEGH,AE,AD}Tm ph ti thiu ca R.p n : {CB,ABG,ABH,AE,AD}Bi Tp 4:Cho R(A,B,C,D,E) v F={AB,BC,AC,BDE,AE,AD}Tm ph ti thiu ca R.p n : AB,BC,BD,BEBi Tp 5:Cho U(ABCDEGH) v F={ABC,BEG,ED,DG,AB,AGBC}Tm ph ti thiu ca U.p n : {AB,AC,DG,ED}

Bi Tp 6:Tm ph ti thiu ca tp ph thuc hm T sau y:T={ABHCK,AD,CE,BGHF,FAD,EF,BHE}p n : {BHC,BHK,AD,FA,EF,BHE}

Bi Tp 7:ChoR(ABCDEGHIJ)F={ABDE,DEG,HJ,JHI,EDG,BCGH,HGJ,EG}Tm ph ti thiu ca G.p n : {AB,BCH,AE,BCG,HJ,JH,JI,ED,EG}

Bi Tp 8:Cho lc quan h = (U,F) viU = ABCDEGHF={ BC ADE, AC BDG, BE AC, CD BH, BH ACG}Tm ph ti thiu ca U.p n : {BCE,ACD,BEA,BEC,CDB,CDH,BHC,BHG}

Bi Tp 9:Cho lc quan h =(U,F) vi U=ABCDEGHF={ ABCD, EBGH, ACD BG, DAEH}Tm ph ti thiu ca U.p n : {ABCD,EBGH,DAE}Bi Tp 10:Cho lc quan h =(Q,F) vi :Q(A,B,C,D,E,G)F={ABC;CA;BCD;ACDB;DEG;BEC;CGBD;CEAG}p n :{ABC,CA,BCD,DEG,BEC,CGB,CEG}IV. KhaBi Tp 1:Cho lc quan h: =(U,F) V i U=ABCDEGH F={AB CDE, AC BCG, BDG, ACHHE, CG BDE }v K = ACGH

Hi rng K c l kho ca lc hay khng?p n : K khng phi l kha.Bi Tp 2:Cho lc quan h =(U, F) vi U=ABCDEGHKF={ ADHBC, GHBE, DCG, CHK}. Tm kha ?p n : ADHBi Tp 3:Cho lc quan h =(U, F) vi U=ABCDGHF={ ABC, ABD, ABGH, HB}. Tm kha ?p n : AB,AHBi Tp 4:Cho lc quan h =(U, F) vi U=ABCDF={ ABC, ABD, DB, CABD}. Tm kha ?p n : C,AB,ADBi Tp 5:Cho lc quan h = (U, F) vi U=ABCDEGH, F={ ABCADH, ABGAEH, AEDG}Hy tm tt c cc kho ca lc .p n : ABCE,ABCGBi Tp 6:

Cho lc quan h =(U, F) viU=ABCDE , F={ DEA, BC, EAD}

a) Tm cc kho ca lc ? p n :BEb) Tp BCE c phi l kho ca khng? v sao ? Khngc) Tp AD c phi l kho ca khng? v sao ? Khngd) Tp BD c phi l kho ca khng? v sao ? Khng

Bi Tp 7 :

Xc nh kho cu cc lc quan h =(U, F) via) U=ABCEDH vF={ABC, CDE, AHB, BD, AD}p n : AHb) U=ABCDMNPQF={AMNB, BNCM, AP, DM, PCA, DQA}p n :DQc) U=ABCDEGHIJF={AJ, AEH, HE, DEH, AI, ABC, CBD, JE}p n : ABGBi Tp 8:Cho lc quan h = (U,F) vi U = ABCDEGHIK vF = { AEK BEH , EH BD , DG BCD, ABCE DE} a. Tp DEGH c phi l kho ca lc cho hay khng ?b. Hy tm mt kho ca lc trn.c. Hy tm tt c cc thuc tnh khng tham gia vo bt k mt kho no.

Bi Tp 9:Cho lc = (U, F) c U = ABCDE vF = { A BD, AC B, D AB, CD BE, BE A } a. Hi rng tp ACD c l kho ca lc hay khngb. Lc c mt hay nhiu kho , tm cc kha?Bi Tp 10:Cho lc quan h = (U, F) vi U = ABCDEGH vF = { AB CDE , BD CGE , DG ACE, AD CDEH, BCG AEH } Lc cho c mt hay nhiu kho,tm cc kha ?V. Dng chun v chun ha CSDLBi tp 1:Dng k thut bng kim tra php tch sau c mt thng tin khnga) =(U, F) vi U=ABCD, F={AB, ACD}, ={AB, ACD}b) =(U, F) vi U=ABCDE, F={AC, BC, CD, DEC, CEA}, ={AD, AB, BE, CDE}c) Xc nh v gii thch dng chun cao nht ca lc quan h =(U, F) vi U=ABCD, F={AC, DB, CABD}

Bi tp 2:Cho lc quan h =(U, F) viU=ABCDEGHF={CDH, EB, DG, BHE, CHDG, CA }Hi rng php tch =(ABCDE, BCH, CDEGH) c kt ni mt thng tin khng.

Bi tp 3:Cho lc quan h =(U, F) viU=ABCD, F={DB, CA, BACD }Xc nh dng chun cao nht ca lc quan h trn

Bi tp 4:Cho lc quan h =(U, F) viU=ABCD, F={CDB, AC, BACD }Xc nh dng chun cao nht ca lc quan h trn

Bi tp 5:Cho =(u, F) viU=ABCDE vF={AC, BC, AD, DEC, CEA}kim tra tnh kt ni khng mt thng tin i vi php tch ={AD, AB, BE, CDE, AE }

Bi tp 6:Cho =(u, F) viU=ABCDEF vF={ABC, CB, ABDE, FA}kim tra tnh kt ni khng mt thng tin i vi php tch ={BC, AC, ABDE, ABDF }

Bi tp 7:Cho =(u, F) viU=ABCDEGF={DG, CA, CDE, AB}kim tra tnh kt ni khng mt thng tin i vi php tch ={DG, AC, SCE, AB }

Bi tp 8:Cho =(u, F) viU=ABCDE vF={AC, BC, CD, DEC, CEA}kim tra tnh kt ni khng mt thng tin i vi php tch ={AC, CD, BE, BC, AE}

Bi tp 9:Cho (=(U, F) viU=XYZW v tpF={YW, WY, XYZ}Dng chun cao nht ca lc l g?

Bi tp 10:Cho (=(U, F) viU=ABCDEG v tp ph thuc hmF={ ABC, ACE, EGD, ABG } ={DEG, ABDEG }Php tch trn c mt thng tin khng?Hy chng minh mi quan h ch c 2 thuc tnh dng chun BCNF?

Bi tp 11:Xt quan h R(ABCDE) v tp ph thuc hmF={ ABCE, EAB, CD }Hy tm dng chun cao nht ca lc ?

Bi tp 12:Xt quan h R(ABCDEG) v tp ph thuc hmF={ AB, CDG , ACE, DG } Hy tm kho ca lc Hy tm dng chun cao nht ca lc

Bi tp 13:Xt quan h R(ABCD) v tp ph thuc hmF={ ABD, ACBD, BC }Hy tm dng chun cao nht ca lc

Bi tp 14:Cho =(u, F) viU=ABCDEFF={ABC, CB, ABDE, FA}Lc c dng BCNF khng ?