Non Trivial FD

Candidate Key

FD’s that Hold on S

3NF Consider R = {A, B, C, D, E, F, G, H} with a set of FDs F = {CD→A, EC→H, GHB→AB, C→D, EG→A, H→B, BE→CD, EC→B} The candidate keys are: {BEFG, CEFG, EFGH}

F = {CD→A, EC→H, GHB→AB, C→D, EG→A, H→B, BE→CD, EC→B} No, R w.r.t. F is NOT in 3NF, because CD→A violates the 3NF requirements. i.e. CD→A is not trivial FD CD is not a superkey CD is not a key, but A is not part of any key of R either

Binary Decomposition Approach Considering R: Keys ={BEFG, CEFG, EFGH} F = {CD→A, EC→H, GHB→AB, C→D, EG→A, H→B, BE→CD, EC→B} Decomposition #1: CD→A is a violating FD R is decomposed into R1 and R2: R1 (A,C,D): We need to project FDs F onto relation R1: A+ = A C+ = CDA (C →DA) D+ = D AC+ = ACD (AC→D) AD+ = AD CD+ = CDA (CD → A) So, F1 = {C→DA, AC→D, CD→A}

R2 ( B,C,D,E,F,G,H): In general, we should project F onto R2. However, if we look carefully, we can easily see that the only difference between R and R2 is attribute A. Attribute A has never appeared on LHS of any FD. So, removing it won’t make any change in F2. So, F2={EC→H, GHB→B, C→D, H→B, BE→CD, EC→B} FDs that are lost in Decomposition #1 are: Lost = {GHB→A, EG→A} Do we need further decomposition?

Consider R1(A,C,D): F1 = {C→DA, AC→D, CD→A} Since C+=ACD, C is a key. C (in C→DA), CD (in CD→A), and AC (in AC→D) are key/super keys. Therefore, we have no violating FD. (So, we are done with this branch.) Consider R2: F2={EC→H, C→D, H→B, BE→CD, EC→B} Keys of R2 = Keys of R = {BEFG, CEFG, EFGH} EC→H is not a violating FD, since H is part of a key. C→D is a violating FD, since C is not a super key and D is not part of any key. So, further decomposition is needed.

Decomposition #2: C→D is a violating FD R2 is decomposed into R21 and R22: R21 (C, D): We need to project F2 onto relation R21: C+=CD D+=D So, F21={C→D} R22 (B,C,E,F,G,H): In general, we should project F2 onto R22. However, if we look carefully, we can easily see that the only difference between R2 and R22 is attribute D. Attribute D has never appeared on LHS of any FD. So, removing it won’t make any change in F22. So, F22 = {EC→H, H→B, BE→C, EC→B}

FDs that are lost in Decomposition #2 is: Lost = {BE→D} So, overall, we’ve lost the following FDs: Lost = {GHB→A, EG→A, BE→D} Do we need further decomposition? Consider R21(C,D): F21={C→D} Since C+=CD, C is a key. Therefore, we have no violating FD. (So, we are done with this branch.) Consider R22(B,C,E,F,G,H): Keys of R22 = Keys of R2 = {BEFG, CEFG, EFGH}

F22 = {EC→H, H→B, BE→C, EC→B} EC→H is not a violating FD since H is part of a key. H→B is not a violating FD since B is part of a key. BE→C is not a violating FD since C is part of a key. EC→B is not a violating FD since B is part of a key. So, we are done with this branch. Overall, we have: R1 (A, C, D)F1 = {C→DA, AC→D, CD→A} R21 (C, D)F21 = {C→D} R22 (B, C, E, F, G, H)F22 = {EC→H, H→B, BE→C, EC→B}

Since R1 includes R21 we might want to remove R21. This is a loss-less join decomposition, but it is not dependency preserving. To make the decomposition dependency preserving, we need to add the lost FDs as new relations. The lost FDs are: Lost = {GHB→AB, EG→A, BE→D} So, we add three relations: L1(A, B, G, H) FL1 = {GHB→AB} L2(A, E, G)FL2 = {EG→A} L3(B, D, E)FL3 = {BE→D}

Synthesis Approach R = {A, B, C, D, E, F, G, H} with a set of FDs F = {CD→A, EC→H, GHB→AB, C→D, EG→A, H→B, BE→CD, EC→B} The candidate keys are {BEFG, CEFG, EFGH} Canonical cover for F is: FC = {C→AD, EC→H, GH→A, EG→A, H→B, BE→C}

Now, we create the relations: R1 = {A, C, D} F1 = {C→AD} R2 = {E, C, H} F2 = {EC→H} R3 = {A, G, H} F3 = {GH→A} R4 = {A, E, G} F4 = {EG→A} R5 = {B, H} F5 = {H→B} R6 = {B, C, E} F6 = {BE→C} Now, we need to check if at least one of the keys exists in the above relations. The candidate keys are {BEFG, CEFG, EFGH}

Since none of these keys is in the relations, this decomposition is not lossless. So, we need to add an extra relation containing those attributes that form any key of R: R7 = {B, E, F, G}F7 = { }

Check lossless join Check if the decomposition of R(A, B, C, D, E, F, G) with the set of FDs F={C→AD, E→G, FG→A, EF→A, G→B, BE→C} into the following relations is lossless join. R1 = {A, C, D} R2 = {E, C, G} R3 = {A, F, G} R4 = {A, E, F} R5 = {B, G} R6 = {B, C, E}

Step 1- Table initialization

Round 1

Round 2