The Boyce - Codd normal form BCNF is an essential normal form for relation schemes in the relational database.. Keys and minimal keys are the important concepts of the relational datamod
Trang 1TI!-p chI Tin hgc va fJieu khien hoc, T 16, S 1 (2000), 15-17
VU DUC THI, LUONG CAO SON
Abstract The Boyce - Codd normal form (BCNF) is an essential normal form for relation schemes
in the relational database This normal form has been used in designing database systems Keys and minimal keys are the important concepts of the relational datamodel The set of minimal keys of relation scheme is Sperner system In this paper we show a new necessary and sufficient conditions for
an arbitrary relation scheme is in BCNF and its set of minimal keys is a given Sperner system
Now we start with some necessary definitions, and in the next sections we formulate our results Definition 1 Let R = {hI, ,hn} be a relation over U, and A, B ~ U Then we say that B
functionally depends on A in R (denoted A L , B) iff
R (Vhi' hj ER)( V a EA)(hda) =h j (a)) ~ (Vb EB)(hdb) =hj(b)).
Let FR ={(A, B) ; A, B ~ U, A L; B}. FR is called the full family of functional depen:dencies
R
of R Where we write (A, B) or A - +B for AI / R B when R, I are clear from the context.
Definition 2 A functional dependency (FD) over U is a statement of the form A - + B , where
A, B ~ U The FD A - + B holds in a relation R if A .L; B We alsosay that R satisfies the FD
R
A- + B
Definition 3 Let U be a finite set, and denote~ P(U) its power set Let Y ~ P(U) x P(U). We say
that Y is an I-family over U iff for all A, B, C , D ~ U
(1) (A, A) EY ,
(2J ' (A , B) EY, (B , C) EY ~ (A, C) EY ,
(3) (A, B) E Y, A ~ C, D ~ B ~ (C, D) E Y,
(4) (A, B) EY, (C, D) EY ~ (A UC, BUD) EY
Clearly, FR is an I-family over U.
It is known [1] that if Y is an arbitrary I-family, then there is a relation Rover U such that
F =Y
Definition 4 A relation scheme S is11 pair (U, F) , where U is a set of attributes, and F is a set of
FDs over U Let F+ be a set of all FDs that can be derived from F by the rules in Definition 3.
Clearly, in [1] if S = (U, F) is a relation scheme, then there is a relation Rover U such that
F R = F+. Such a relation is called an Armstrong relation of S
Definition 5 Let R be a relation over U, S =(U, F) be a relation scheme, Y be an I-family over U,
andA ~ U Then A is a key of R (a key of S, a key of Y) if A L , U (A - + U E F+, (A, U) E Y).
R
A is a minimal key of R(S, Y) if A is a key of R(S, Y) and any proper subset of A is not a key of R(S , Y) Denote K R, (K s, Ky) the set of all minimal keys of R(S , Y)
Clearly, K R, K s, K y are Sperner systems o er U
Definition 6 Let K be a Sperner system over U. We define the set of antikeys of K, denote by
K- I as follows:
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It is easy to se that K - 1 is also a Sperner system over U.
of two attributes to be the elementary step of algorithms Thus, if we assume that subsets of U are
not in M, since it is the intersection of the empty collection of sets
Denote N = {A EI: A = 1= n {A ' EI A CA ' }}.
Definition 8 Let R be a relation over U, and ER the equality set of R , i.e ER ={Eij : 1 :Si<j :S
we presented the following result
x ; ' =T•.
form ifA- + {a} f/- F + for A+ = 1 = R , af/ - A
If arelation scheme is changed to a relation we have the defini on of BCNF for relation
2 RESULTS
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Z ~~)- According to defi n i o n o f BCN F , i f s is in BC!-~!? then for e v e y B E F { ~~ B - G E Z( 5 ).
Cor.sequcnt.lv, we obtain AI" t ;; ; Z (s).
Assume that We have C.). By Proposition 1 and ac ording to definitions of Z(s), T(K-l )
[0 "rop05ition 1 there exists a B E K; 1such that A+ t ;; ; B. Cie rly a E B and A t;; ; B - a hold
Cor.ccqueri ly, we have (B - a ) + = B. This is a co tadictio Thus, s is in BCl'iF The proof i s
complete
Clearly, the right-han side and the left-hand side of (*) don't depend on s.
holds
is obvious
(B - a )+ = B - a.
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