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Date: Tue, 12 Aug 2008 01:50:58 -0700 (PDT)
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On Aug 8, 4:11 pm, "Brian Selzer" <> wrote:
> <> wrote in message
> > On Aug 2, 1:10 pm, "Brian Selzer" <> wrote:
> > > <> wrote in message
> > >
> > > > Hi all,
> > > > BCNF
> > > > the following is the definition is the definition of BCNF , which i
> > > > saw in a schaum series book
> > > > 1) The relation is 1 N.F
> > > > 2) for every functional dependency of the form X -> A , we have
> > > > either A C X or X is a super key of r. in other words,
> > > > every functional dependency is either a trivial dependency or in
> > > > the case that the functional dependency is not trivial then X must
> > > > be a super key.
> > > > now my questions are as follows
> > > > 1)
> > > > we know that 2-ND normal form is all about separating partial
> > > > dependencies and full dependencies.third normal form is all about
> > > > removing transitive dependencies, in these lines can any one give
> > > > simple/ easy to understand method/explanation for converting a
> > > > relation in 3rd normal form to BCNF
> > > A relation schema is in 3NF iff for every functional dependency the
> > > determinant is a superkey or the dependent is prime; a relation schema
> > > is in
> > > BCNF iff every determinant is a superkey. A schema that is in 3NF but
> > > not
> > > in BCNF will have one or more determinants that are not superkeys. Find
> > > them and eliminate them.
> > > > 2) how correct is the following definition of transitive
> > > > dependencies
> > > > transitive dependencies
> > > > assume that A,B, and C are the set of attributes of a relation(R).
> > > > further assume that the following
> > > > functional dependencies are satisfied simultaneously : A -> B , B -/-
> > > >> A, B -> C , and C -/-> A and A -> C
> > > > observe that C -> B is neither prohibited nor required. if all these
> > > > conditions are true, we will say that attribute C is transitively
> > > > dependent on attribute on A
> > > It is not correct: what if B = C or C is a subset of B?
> > this is what the the reply i got from the authors of the book
> > The condition for having a transitive dependency is as follows:
> > A -> B   B-> C from this you will infer that A ->C provided that B
> > does not determine A and C does not determine A. The reason for
> > requiring that B does not determine A is because we are assuming that
> > A is the only key. If B were to determine A then B will also be a key.
> > Same thing applies to C. Remember that you are going from 1NF to 2NF.
> > This latter form requires that you do not have any partial
> > dependencies on any primary or candidate key. That is no subset of A
> > can determine B if A is a key or candidate key.
> Bunk!  Suppose that R has attributes {V, W, X, Y},
> that A is the set of attributes {V, W, X, Y},
> that B is the set of attributes {W, X, Y}
> and that C is the set of attributes {X, Y}.
> R is in 2NF.
> Clearly A --> B, B -/-> A, B --> C, A --> C, and C -/-> A,
> but C is not /transitively/ dependent on A: C is /trivially/ dependent on A.
> > If B=C then they are
> > the same attribute and every attribute determines itself. That is what
> > the reflexivity axiom is all about. Remember that 2NF requires that
> > there cannot be partial dependencies on the key. It does not talk
> > about partial dependencies among the other attributes if these
> > attribute are not primary key or candidate key. Just make sure that
> > before you guarantee that a set of relationships are in 2NF you find
> > ALL keys. From this set you will choose a primary key. There got to be
> > only one primary key. The other possible keys are candidate. Tell your
> > friends that they are not correct the definition in the book is the
> > original and only definition that there is about transitivity.
> Assuming that you copied the definition from the book word for word, the
> definition in the book is wrong because it does not exclude trivial
> functional dependencies.  Here is a definition that appears in a paper by
> Carlo Zaniolo, "A New Normal Form for the Design of Relational Database
> Schemata", ACM Transactions on Database Systems, Vol. 7, No. 3, September
> 1982:
> Let R be a relation with attribute set U, let X be a subset of U (not
> necessarily proper), and let A be a member of U.  A is /transitively/
> dependent on X if there exists a Y which is a subset of U (not necessarily
> proper) such that X --> Y, Y -/-> X, Y --> A, and A is not a member of Y.
> [Zaniolo's definition used symbols instead of 'be a subset of ... (not
> necessarily proper)' and 'is/is not a member of'.  Sometimes I've found that
> the symbols for 'is a subset of' and 'is a proper subset of' are not used
> consistently.  This may be one case of that because clearly, in order for
> there to be a transitive dependency, both X and Y must both be /proper/
> subsets of U, since A can't be a member of Y, so if Y were a subset of X
> which would be the case when X = U, then A couldn't be a member of X.]
> Note that the final condition, 'A is not a member of Y,' excludes the
> trivial case.
> > Just
> > remember that in the book we are assuming that A is the PK. If as your
> > friends say B -> A and A is a key then B is a key. As I mentioned
> > before this applies also to C. If A, B, and C were all keys then the
> > definition of 2NF and 3NF are moot because there cannot be partial or
> > transitive dependencies because these definitions require that there
> > exist other attributes that are not primary or candidate keys. You
> > need to be careful when you go to 3NF if you have PK that overlap.
> > Then you have to move to BC normal form.
> > Thanks for reading the book and reading it carefully. If you find any
> > mistakes please let me know, Pauline and I are in the process of
> > getting ready for the  2nd. Ed.
> > Ramon A. Mata-Toledo

this again is the reply i got from authors of the book

I am glad that you are taking databases seriously. With regard to your question. Assume you have the relation R with the attributes that you have defined. The relation R is in 2NF if, first, it is in 1NF. We can assume that none of the attributes V, W, X, and Y have attributes of their own. Therefore, the relation is in 1NF. Now, the 2NF requires that you have a key but it also assume that there are non prime attributes. These non prime attributes cannot be partially dependent on the key. Yes, you have a key. It is obvious from the reflexivity axiom that the key determines all other attributes. In fact, {V, W, X,

Y} -> V; {V, W, X, Y}-> W; {V, W, X, Y} -> X; {V, W, X, Y} ->Y. You
also have that {V, W, X, Y} -> {W,X,Y} by the axiom of augmentation.
Finally, {V, W, X, Y} -> {X,Y} by the same axiom. The apparent
contradiction that you have found to the definition of 3NFcomes from not having nonprime attributes which are not partially dependent upon ANY key. Take into account also that the def of 3NF requires that the relation be previously in 2NF (no nonprime attribute can be partially dependent upon any key).Remember the definitions (2NF, 3NF) assume that the key A determines every other attribute but it also assumes the existence of nonprime attributes which are not partially dependent upon the key. All the attributes that you have here, B and C are partially dependent upon the key. In the def of transitive dependence (refer to the graph in the book) A is assumed to be a single key. B is a nonprime attribute and it cannot be partially dependent upon the key. Finally, C must also be a nonprime attribute. In case, you forgot, nonprime attributes are those attributes that do no participate in any key.

If there are no partial dependencies you are in 2NF. If you do not have transitive dependencies you are in 3NF. Obviously, I am assuming that we have a key for the relation as determined by the set of given FDs. Finally, if you have a relation with attributes A, B, and C and they all can be used as key and they do not overlap then you are already in 3NF.

By the way, in the book all attributes A,B, and C are assumed to be simple attributes unless defined otherwise. This is to simplify the explanations.

If you want to take a more formal book, take a look at The Theory of Relational Databases by David Maier. It is heavy in mathematics.

Before, I forgot, Trivial FDs are those that are determined by the Axiom of Reflexivity. A comment about the use of notations. In the DB world, there are definitions and definitions. Zaniolo is top expert in DB but if you read some of his papers you will find that the same concept may have more than one interpretation. This is typical of DB. That is, why it is very important to know what definitions you are using. Take for example, the word set, I believe that the last time I checked there were more than 200 definitions of this word in different DBMSs alone. When working with FD and non redundant FDs you need to exclude trivial FDs.

The Definition that I used in the book is the same as Maier which in turn is the definition that Date uses. I do a lot of consulting in DB across the world in ORACLE. It is fun to work with set of FDs developed by the designers who are using the IBM concepts and notations and sometimes notations and definitions of their own. Take into account that the same thing happens sometimes in mathematics. One author considers the set of natural numbers beginning with one. Others may assume that it includes zero. That is, sometimes you need to check what the definitions are to be consistent. Hope this helps ;if not I hope to hear from you again soon.

I have enjoyed the discussion across the ocean. Be well, Received on Tue Aug 12 2008 - 10:50:58 CEST

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