Karlstad University
Department of Information Systems
Adapted for a textbook by Date C. J.
An Introduction to
Database Systems
Pearson Addison Wesley, 2004
Database Design
Remigijus GUSTAS
Phone: +46-54 700 17 65
E-mail: Remigijus.Gustas@kau.se
http://www.cs.kau.se/~gustas/
Functional Dependence
Functional dependence is a many to one
relationship from one set of attributes to another
within a given relvar
Let r be a relation, and let X and Y be subsets of
the attributes of r
X → Y says
“Y is functionally dependent on X”, or
“X functionally determines Y”
X is the determinant; Y the dependent
An FD is trivial if and only if the right side is a
subset of the left side (not interesting in practice)
{ S#, P# } → S#
All other dependencies are called nontrivial
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Functional Dependency
A particular dependency between two attributes. For a
given relation, attribute B is functionally dependent on attribute
A is, for every valid value of A, that value of A uniquely
determines the value of B
EMP# ENAME
EMP# {ENAME, DEPT#, SALARY}
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EMP#
ENAME
DEPT#
SALARY
E1
Lopez
D1
40K
E2
Cheng
D1
42K
E3
Finzi
D2
30K
E4
Saito
D2
35K
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Armstrong’s Axioms
The set of all FDs that are implied by a given set
S of FDs is called the closure of S
1.
2.
3.
4.
5.
6.
7.
Reflexivity: If B is a subset of A then A → B
Augmentation: If A → B, then AC → BC
Transitivity: If A → B and B → C, then A → C
Self-determination: A → A
Decomposition: If A → BC, then A → B and A → C
Union: If A → B and A → C, then A → BC
Composition: If A → B and C → D, then AC → BD
A, B, and C are subsets of relvar R, and
AB signify the union of A and B
Three first rules are sound (all FDs) and complete (no
additional FDs)
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Irreducible Sets of Dependencies
Given some particular set S of FDs that
need to be enforced, it is sufficient for the
database system to enforce the FDs in
irreducible equivalent
Let S1 and S2 be two sets of FDs. If every FD
implied by S1 is implied by S2, then S2 is a cover
of S1
A set S of FDs is irreducible iff:
The right side of every FD in S involves one attribute (a
singleton set)
The left side is irreducible
No FD in S can be discarded without changing the
closure of S
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Normalized Relations
Redundancies
removed by
breaking into
two separate
relations
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Nonloss (Information-Preserving)
Decomposition
Normalization uses a process of projection
to decompose relvars
Recomposition is a process of joins
The decomposition of relvar R into
projections R1…Rn is nonloss if R = the join
of R1…Rn
The normalization procedure can be seen
as a method for eliminating functional
dependencies that do not emanate from a
candidate key
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Normalization
- the process of converting complex data
structures into more simple, stable data
structures
First Normal From (1NF)
Unique rows
All attributes are atomic
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Normalization (cont.)
Second Normal Form (2NF)
Each nonprimary key attribute is identified by the
whole key (called full functional dependency).
Third Normal Form (3NF)
Nonprimary key attributes do not depend on
each other (i.e. no transitive dependencies).
The result of normalization is that every
nonprimary key attribute depends upon the
whole primary key.
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1NF but not 2NF
EMPLOYEE2(Emp_ID, Name, Dept, Salary, Course, Date_Completed)
Functional dependencies:
1. Emp_ID Name, Dept, Salary
2. Emp_ID, Course Date_Completed
partial key dependency
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2NF (actually, also 3NF)
EMPLOYEE1(Emp_ID, Name, Dept, Salary)
Functional dependencies:
Emp_ID Name, Dept, Salary
EMPCOURSE(Emp_ID, Course,
Date_Completed)
Functional dependency:
Emp_ID, Course Date_Completed
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First Normal Form
A relvar is in 1NF if and only if in every legal
value of that relvar, every tuple contains
exactly one value for each attribute
In this way, relvars are always in 1NF
A relvar in 1NF may display functional
dependencies other than those emanating
from the primary key
Such non-primary-key dependencies
promote update anomalies
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Update Anomalies
“Update anomalies” include three operations:
An INSERT anomaly occurs when the user
wishes to record a subordinate fact that is not
dependent on the primary key
e.g., recording a supplier location before the supplier
supplies a part
A DELETE anomaly
conversely, may delete the location inadvertently
An UPDATE anomaly occurs when many updates
are required to record a simple fact
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Second Normal Form
A relvar is in 2NF if and only if it is in 1NF and
every nonkey attribute is irreducibly dependent on
the primary key
Assumes only one candidate key
A relvar in 2NF is less susceptible to update
anomalies, but may still exhibit transitive
dependencies
Both attributes in a transitive dependency are
irreducibly implied by the primary key, and each
implies the other
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2NF but not 3NF
SALES(Customer_ID, Customer_Name, SalesPerson, Region)
Functional dependencies:
1. Customer_ID Customer_Name, SalesPerson, Region
2. SalesPerson Region
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transitive
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Third Normal Form
A relvar is in 3NF if and only if the nonkey
attributes are both
mutually independent and
irreducibly dependent on the primary key
(informal definition) A relvar is in 3NF if and
only if, for all time, each tuple consists of a
primary key value that identifies some
entity, together with a set of zero or more
mutually independent attribute values that
describe that entity in some way
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Converted to 3NF
SALES1(Customer_ID, Customer_Name,
SalesPerson)
Functional dependencies:
Customer_ID Customer_Name,
SalesPerson
SPERSON(SalesPerson, Region)
Functional dependency:
SalesPerson Region
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Dependency Preservation
Dependency preservation refers to a
specific case of nonloss decomposition,
such that the normalized relvars are
independent of each other
Some nonloss decompositions do not
exhibit dependency preservation
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Customer_ID {Customer_Name, SalesPerson}
SalesPerson Region
Constraint Customer_ID Region will be enforced
automatically if the other two constraints are enforced
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Dependency Preservation: Example
Dependencies are preserved in this projection:
Customer {Customer_ID, Customer_Name,
SalesPerson}
CS {SalesPerson, Region}
Dependencies are not preserved in this one:
SC {Customer_ID, Customer_Name,
SalesPerson}
CS {Customer_ID, Region}
Although the second is nonloss, you still cannot
update them independently
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Boyce/Codd Normal Form
BCNF refers to decompositions involving
relvars with more than one candidate key,
where the candidate keys are composite
and overlapping
A relvar is in BCNF if and only if every
nontrivial, left- irreducible FD has a
candidate key as its determinant
That is, a relvar is in BCNF if and only if
every determinant is a candidate key
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Boyce/Codd Normal Form: Example
{S,J} T
TJ
S
J
T
Smith
Math
Prof White
Smith
Physics
Prof Green
Jones
Math
Prof White
Jones
Physics
Prof Green
ST (S,T), TJ (T,J)
Two projections are in BCNF,
but the trouble is that they are not independent.
Decomposition does not in fact avoid anomalies.
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