We have the attributes: {A, B, C, D, E, F, G}.Consider the following functional dependenciesB, E -> C, D, FA, G -> B, CA, F -> B, DF, G -> ED, E -> A, B, GD, E -> A, BThe first step in computing the normal form, is to derive a canonical (minimal) set of functional dependencies. We have already done this first step for you. The following is a set of canonical functional dependencies (for the dependencies shown above):B, E -> DD, E -> AA, F -> DA, F -> BA, G -> BB, E -> FF, G -> EA, G -> CD, E -> GApply the synthesis algorithm to derive a Boyce-Codd normal form for the relation R(A, B, C, D, E, F, G).Write your answer as followsFunctional dependencies: write one dependency per line, for exampleA,B -> CB,C,D -> A,FSplitting steps: write one split per line, for examplespliting R(A,B,C,D) into S(A,B,C) and T(A,D) -- the relation names are not importantspliting (A,B,C) into (A,B) and (A,C) -- and can even be omitted...
Question
We have the attributes: {A, B, C, D, E, F, G}.Consider the following functional dependenciesB, E -> C, D, FA, G -> B, CA, F -> B, DF, G -> ED, E -> A, B, GD, E -> A, BThe first step in computing the normal form, is to derive a canonical (minimal) set of functional dependencies. We have already done this first step for you. The following is a set of canonical functional dependencies (for the dependencies shown above):B, E -> DD, E -> AA, F -> DA, F -> BA, G -> BB, E -> FF, G -> EA, G -> CD, E -> GApply the synthesis algorithm to derive a Boyce-Codd normal form for the relation R(A, B, C, D, E, F, G).Write your answer as followsFunctional dependencies: write one dependency per line, for exampleA,B -> CB,C,D -> A,FSplitting steps: write one split per line, for examplespliting R(A,B,C,D) into S(A,B,C) and T(A,D) -- the relation names are not importantspliting (A,B,C) into (A,B) and (A,C) -- and can even be omitted...
Solution
The synthesis algorithm for deriving a Boyce-Codd normal form (BCNF) involves the following steps:
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Find a canonical cover for F (set of functional dependencies). This has already been done for you.
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For each functional dependency in the canonical cover, create a new relation with the attributes of that functional dependency.
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If no relation contains a candidate key for R, create one more relation schema and include this candidate key.
Applying these steps to your given set of canonical functional dependencies, we get:
Functional dependencies:
B, E -> D
D, E -> A
A, F -> D
A, F -> B
A, G -> B
B, E -> F
F, G -> E
A, G -> C
D, E -> G
Splitting steps:
Splitting R(A, B, C, D, E, F, G) into R1(B, E, D), R2(D, E, A), R3(A, F, D, B), R4(A, G, B), R5(B, E, F), R6(F, G, E), R7(A, G, C), and R8(D, E, G).
Please note that the relation names are not important and can even be omitted. The important part is to ensure that each new relation satisfies BCNF.
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1. Normal formsWe have the attributes: {A, B, C, D, E, F, G}.Consider the following functional dependenciesA, C, F -> D, EA -> CA, C, F -> B, EF -> A, BA, C -> E, F, GA, F, G -> C, DB, C -> DB -> AThe minimal keys are:{B}{F}{A}Determine whether these functional dependencies are in the following normal form(s):Boyce Codd normal form2. Normal formsWe have the attributes: {A, B, C, D, E, F, G}.Consider the following functional dependenciesF -> E, GA, G -> C, D, EA, B -> FD -> F, GB, E, F -> D, GA, D, E -> B, FB, D, F -> A, EF, G -> B, D, EThe minimal keys are:{D}{F}{A, B}{A, G}Determine whether these functional dependencies are in the following normal form(s):Boyce Codd normal form3. Normal formsWe have the attributes: {A, B, C, D, E, F, G}.Consider the following functional dependenciesB, C, D -> E, GB -> A, ED, E -> CA, C -> B, D, EB -> C, DD, F, G -> AG -> BB, C -> FThe minimal keys are:{B}{G}{A, C}{A, D, E}Determine whether these functional dependencies are in the following normal form(s):Boyce Codd normal form4. Normal formsWe have the attributes: {A, B, C, D, E, F, G}.Consider the following functional dependenciesA, F, G -> CC, F -> GB -> C, FA, D, E -> CE, F, G -> B, CA, B, F -> D, E, GB, C -> A, DA, C, F -> GThe minimal keys are:{B}{C, E, F}{E, F, G}{A, D, E, F}Determine whether these functional dependencies are in the following normal form(s):Boyce Codd normal form5. Normal formsWe have the attributes: {A, B, C, D, E, F, G}.Consider the following functional dependenciesD -> BF -> AB, C, D -> E, FA -> GE, F -> CA, D -> E, FE, F -> GC, F -> A, EThe minimal keys are:{D, F}{C, D}{A, D}Determine whether these functional dependencies are in the following normal form(s):Boyce Codd normal form6. Normal formsWe have the attributes: {A, B, C, D, E, F, G}.Consider the following functional dependenciesC -> EE -> D, GD -> FF -> B, GF -> C, D, EG -> FC -> FC, G -> EThe minimal keys are:{A, E}{A, C}{A, F}{A, G}{A, D}Determine whether these functional dependencies are in the following normal form(s):Boyce Codd normal form
Functional dependencies: implicationWe have the attributes: {A, B, C, D, E, F, G}.Consider the following functional dependenciesA, D, F -> EE, F, G -> DA, B -> FE -> BC, G -> D, FC, G -> B, EA, C -> EA -> C, EMark all functional dependencies that are implied by the dependencies above:A, C -> EA, C -> BA, C -> B, DA, C -> CA, C -> C, F
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