Rensselaer Polytechnic Institute
CSCI-4380 – Database Systems
David Goldschmidt, Ph.D.

Given a set of attributes X, the
closure X+ is the set of attributes
functionally determined by X

Given a relation R and a set F of functional
dependencies, we need a way to find
whether a functional dependency X  Y is
true with respect to F

Given relation R with attributes A, B, C, D, E
and A  BC, CD  E, BE  C
 AE  _____?
 From reflexivity, AE+ = { A, E }
 From A  BC, AE+ = { A, B, C, E }
 No other rules are applicable or add to AE+
 We conclude that AE  ABCE or simply AE  BC
 Or AE  A, AE  B, AE  C, and AE  E

Given a set F of functional dependencies,
the closure X+ of a set of attributes X is
determined by the following algorithm:
 Initialize X+ to X
 Repeat until X+ does not change:
▪ Find any unapplied functional dependency Y  Z
in F such that Y  X+
▪ Set X+ = X+  Z

A key K for a given relation R is a minimal set
of attributes A1, A2, ..., An such that closure
{A1, A2, ..., An}+ is the set of all attributes of R
 MusicGroup(name, artist, genre, dateformed,
datefirstjoined)
 name  genre dateformed
 name artist  datefirstjoined
 K must be (name, artist) because K+ = {name,
artist, genre, dateformed, datefirstjoined}

Given a set F of functional dependencies,
closure F+ is the set of all functional
dependencies implied by F
 F+ can be found using the set of inference rules
(reflexivity, transitivity, augmentation, etc.)

Sets F1 and F2 of functional dependencies
are considered equal if they have the same
closure (i.e. F1+ = F2+)

In addition to determining closure F+ for
set F of functional dependencies, we can
also derive any functional dependency that
follows from F via Armstrong’s axioms:
 Reflexivity (if Y  X, then X  Y)
 Augmentation (if X  Y, then XZ  YZ)
 Transitivity (if X  Y and Y  Z, then X  Z)

Given a set F of functional dependencies,
any set of functional dependencies that’s
equivalent to F is called a basis
 We limit the possibilities by requiring
that each dependency has a single
attribute on the right-hand side
 How many bases are there for a relation R
with n functional dependencies in F?

A minimal basis for a relation R is a basis B
that satisfies the following conditions:
 All functional dependencies in B have
singleton right-hand sides
 If any functional dependency is removed from B,
the result is no longer a basis
 If any left-hand side attribute is removed from
a functional dependency of B, the result is no
longer a basis

Given basis B, we can determine whether it
is a minimal basis via the algorithm below:
 For each functional dependency X  Y in B,
check if B – { X  Y } still implies X  Y
▪ if so, remove X  Y
 For each functional dependency XW  Y in B,
check if X+ is the same with respect to F
and ( F – { XW  Y } )  { X  Y }
▪ if so, replace XW  Y with X  Y

Given relation R( A, B, C, D ) and
functional dependencies AB  C,
C  D, and D  A
 What are the keys of R?
 What are the superkeys of R that are not keys?
 Is the given set of functional dependencies a
basis? Is it a minimal basis?

What’s wrong with the relation below?
 MusicGroup( name, artist, genre,
dateformed, datefirstjoined )
 i.e. how can tuples
become corrupted
or incorrect?

Without normalization, problems with
relations include:
 Unnecessary redundancy
 Insert anomalies
 Update anomalies
 Delete anomalies

Splitting a relation into two (more specific)
relations is called decomposition
 The objective is to have each
resulting relation be atomic
 i.e. each relation should contain
only information related to the key

A given relation R with set F of functional
dependencies is in BCNF if and only if all
functional dependencies X  Y in F are:
 either trivial (i.e. Y  X)
 or X is a superkey of R

If relation R is not in BCNF, it is possible to
use decomposition to transform R to BCNF

Given a set F of functional dependencies for
relation R( A1, A2, ..., An ) that is not in BCNF:
 Convert F to a minimal basis
 Find an X  Y that violates BCNF
 Compute closure X+
 Decompose R into:
▪ R1 containing all attributes of X+
▪ R2 containing { A1, A2, ..., An } – ( X+ – X )
 Project functional dependencies onto R1 and R2

Given relation R( A, B, C, D, E ) and
functional dependencies AB  AC,
CE  D, B  A, and D  AE
 What are the keys of R?
 What are the superkeys of R that are not keys?
 Is the given set of functional dependencies a
basis? Is it a minimal basis?
 Is relation R in BCNF? If not, decompose R such
that it is in BCNF