# FUNCTIONAL DEPENDENCIES by 4YqikvN

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```									FUNCTIONAL DEPENDENCIES
Definition
• Let R be the relation, and let x and y be the
arbitrary subset of the set of attributes of R. Then
we say that Y is functionally dependent on x – in
symbol.
XY
(Read x functionally determines y) –
If and only if each x value in R has associated with it
precisely one y value in R
In other words
Whenever two tuples of R agree on their x value,
they also agree on their Y value.
Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Example (SCP Relation)
S#       City                      P#             QTY

S1       London                    P1             100

S1       London                    P2             100

S2       Paris                     P1             200

S2       Paris                     P2             200

S3       Delhi                     P2             300

S4       Kolkata                   P2             400

S4       Kolkata                   P2             400

S4       Kolkata                   P5             400

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Example (SCP Relation) (contd..)
One FD : - ( { S#}  {City})

• Because every tuple of that relation with a
given S# value also has the same city value.

• The left and right hand side of an FD are
sometimes called determinant and the
dependents respectively.

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Exercise
Check whether following relation satisfy FD
as not
• < S#, P# >  <QTY>
• <S#, P#>  <City>
• < S#, P#>  <City, QTY>
• <S#, P#>  <S#>
• <S#, P#>  <S#, P#, QTY, City>
• <OTY>  <S#>

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Extended definition over basic one
• Let R be the relation variable, and let x and y be
arbitrary subset of the set of attributes of R. Then
we says that Y is functionally dependent on x – in
symbol.
XY
• If and only if, in every possible legal value of R,
each x value has associated with it precisely one y
value
Or in other words
• In every possible legal value of R, whenever two
tuple agree on their X values, they also agree on
their Y value.
Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
TRIVIAL & NON-TRIVIAL DEPENDENCIES

• One-way to reduce the size of the set of FD
we need to deal with is to eliminate the
trivial dependencies.
• An FD is trivial if and only if the right hand
side is a subset of the left hand side.
e.g. <S#, P#>  <S#>. (Trivial)
• Nontrivial dependencies are the one, which
are not trivial.

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
CLOSURE of a set of dependencies

• The set of all FDs that are implied by a
given set S of FDs is called the closure of S,
denoted by S+

• So we need an algorithm which compute S+
from S.

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Algorithm
CLOSURE (Z, S): = Z;
DO “ forever”
For each FD X -> Y in S
Do;
if X < CLOSURE [Z, S] /* <= “subset of” */
then CLOSURE [Z,S] : = CLOSURE [Z, Z] U Y;
end;
If CLOSURE [Z, S] did not change on this iteration.
Then leave loop; /* Computation complete */
End;
Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Example
Suppose use are given R with attributes
A, B, C, D, E, F, and FDs
• A  BC
• E  CF
• BE
• CD  EF
Then compute the closure (A, B)+ of the set of
attributes under this set of FD’s

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Solution
1. We initialize the result CLOSURE [Z, S]
to <A, B>
2. We now go round the inner loop four
times, once for each for the given FDs. An
the first iteration (For FD A  BC), we
find that the left hand side is indeed a
subset of CLOSURE (Z, S) as computed
so for, so we add attributes (B and C) to
the result. CLOSURE [Z, S] is now the set
<A, B, C>.
Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Solution
3. On the second iteration (for FD E  CF>.
we find that the left hand side is not a
subset of the result as computed so for,
which than remain unchanged.
4. On the third iteration (For FD B E), we
add E to the closure, which now has the
value <A, B, C, E>
5. On the fourth iteration, (for FD CD 
EF), remains unchanged.

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Solution

6. Inner loop times, on the first iteration no
change, second, it expands to <A,B, C, E,
F> third & fourth, no change.

7. Again inner loop four times, no change,
and so the whole process terminates.

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Armstrong rules
1. Reflexivity: if B is a subset of A, then A  B.
2. Augmentation: if A  B then AC  BC
3. Transitivity: it A  B and B  C then A  C.
4. Self – determination: A  A.
5. Decomposition: If A  BC, then AB, AC.
6. Union: it A  B and A  C, then A  BC
7. Composition: if A  B, C  D then AC 
BD.
8. If A  B and C  D, then All (C – B)  BD

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Armstrong rules (contd..)
•   Now we define a set of FD to be
irreducible as minimal; if and only if it
satisfies the following two properties.
(1) The right hand side of every FD in S
involve just one attribute (i.e., it is a
singleton set)
(2) The left hand side of every FD in S is
irreducible in turn meaning that no
attribute can be discarded from the
determinant without changing the
CLUSURE S+.
Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Example

•   A  BC,
•   BC
•   AB
•   AB  C
•   AC  D

Compute an irreducible set of FD that is
equivalent to this given set.
Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Solution
(1) The step is to rewrite the FD such that
each has a singleton right hand side.
•     AB
•     AC
•     BC
•     AB
•     AB  C
•     AC  D
We observe that the FD A  B occurs twice.
So one occurrence will be eliminated.
Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Solution
2. Next, attributed C can be eliminated from
the left hand side of the FD AC  D

•    Because we have A  C,
•    By augmentation A  AC
•    And we are given AC  D,
•    So A  D by transitivity;

Thus C on the left hand side is redundant.
Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Solution
3. Next, we observe that the FD AB  C can be
eliminated, because again we have
AC
By augmentation AB  CB
By decomposition AB  C

4. Finally, the FD A  C is implied by the FD A
 B and B  C, so it can be eliminated.
Now we have A  B
BC
AD
This set is irreducible.
Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Example

•              A  BC
•              BE
•              CD  EF

Show that FD AD  F for R.

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Solution

1.   A  BC (given)
2.   A  C (1, decomposition)
3.   AD  CD (2, augmentation)
4.   CD  EF (given)
5.   AD  EF (3 & 4, transitivity)
6.   AD  F (5, decomposition

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Normalization
Learning Objectives
• Definition of normalization and its purpose
in database design
• Types of normal forms 1NF, 2NF, 3NF,
BCNF, and 4NF
• Transformation from lower normal forms to
higher normal forms
• Design concurrent use of normalization and
E-R modeling are to produce a good
database design
• Usefulness of denormalization to generate
information efficiently
Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Normalization
• Main objective in developing a logical
data model for relational database
systems is to create an accurate
representation of the data, its
relationships, and constraints.

• To achieve this objective, must identify a
suitable set of relations.

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Normalization
• Four most commonly used normal forms are first
(1NF), second (2NF) and third (3NF) normal
forms, and Boyce–Codd normal form (BCNF).

• Based on functional dependencies among the
attributes of a relation.

• A relation can be normalized to a specific form to
prevent possible occurrence of update anomalies.

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Normalization
• Normalization is the process for assigning attributes
to entities
– Reduces data redundancies
– Helps eliminate data anomalies
– Produces controlled redundancies to link tables
• Normalization stages
–   1NF - First normal form
–   2NF - Second normal form
–   3NF - Third normal form
–   4NF - Fourth normal form

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Data Redundancy
• Major aim of relational database design is to
group attributes into relations to minimize
data redundancy and reduce file storage space
required by base relations.

• Problems associated with data redundancy are
illustrated by comparing the following Staff
and Branch relations with the StaffBranch
relation.

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Data Redundancy

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Data Redundancy
• StaffBranch relation has redundant data: details
of a branch are repeated for every member of
staff.

• In contrast, branch information appears only
once for each branch in Branch relation and only
branchNo is repeated in Staff relation, to
represent where each member of staff works.

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Update Anomalies
• Relations that contain redundant
information may potentially suffer from
update anomalies.

• Types of update anomalies include:
– Insertion,
– Deletion,
– Modification.

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Functional Dependency
• Main concept associated with normalization.

• Functional Dependency
– Describes relationship between attributes in a
relation.
– If A and B are attributes of relation R, B is
functionally dependent on A (denoted A  B), if
each value of A in R is associated with exactly one
value of B in R.

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Functional Dependency
• Property of the meaning (or semantics)
of the attributes in a relation.

• Diagrammatic representation:

 Determinant   of a functional dependency refers
to attribute or group of attributes on left-hand
side of the arrow.
Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Example - Functional
Dependency

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Functional Dependency
• Main characteristics of functional
dependencies used in normalization:
– have a 1:1 relationship between attribute(s)
on left and right-hand side of a dependency;
– hold for all time;
– are nontrivial.

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Functional Dependency
• Complete set of functional dependencies for a
given relation can be very large.
• Important to find an approach that can reduce
set to a manageable size.
• Need to identify set of functional dependencies
(X) for a relation that is smaller than complete
set of functional dependencies (Y) for that
relation and has property that every functional
dependency in Y is implied by functional
dependencies in X.
Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
The Process of
Normalization
• Formal technique for analyzing a relation
based on its primary key and functional
dependencies between its attributes.
• Often executed as a series of steps. Each
step corresponds to a specific normal
form, which has known properties.
• As normalization proceeds, relations
become progressively more restricted
(stronger) in format and also less
vulnerable to update anomalies.
Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Relationship Between
Normal Forms

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Unnormalized Form (UNF)
• A table that contains one or more
repeating groups.

• To create an unnormalized table:
– transform data from information source
(e.g. form) into table format with columns
and rows.

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
First Normal Form (1NF)
• A relation in which intersection of each
row and column contains one and only
one value.

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
UNF to 1NF
• Nominate an attribute or group of
attributes to act as the key for the
unnormalized table.

• Identify repeating group(s) in
unnormalized table which repeats for the
key attribute(s).

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
UNF to 1NF

• All key attributes defined
• No repeating groups in table
• All attributes dependent on
primary key

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Second Normal Form (2NF)
• Based on concept of full functional
dependency:
– A and B are attributes of a relation,
– B is fully dependent on A if B is functionally
dependent on A but not on any proper subset of A.

• 2NF - A relation that is in 1NF and every non-
primary-key attribute is fully functionally
dependent on the primary key (no partial
dependency)

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
1NF to 2NF
• Identify primary key for the 1NF relation.

• Identify functional dependencies in the
relation.

• If partial dependencies exist on the
primary key remove them by placing them
in a new relation along with copy of their
determinant.
Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
2NF Conversion Results
Figure 4.5

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Third Normal Form (3NF)
• Based on concept of transitive dependency:
– A, B and C are attributes of a relation such that if A 
B and B  C,
– then C is transitively dependent on A through B.
(Provided that A is not functionally dependent on B or
C).

• 3NF - A relation that is in 1NF and 2NF and in
which no non-primary-key attribute is transitively
dependent on the primary key.

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
2NF to 3NF
• Identify the primary key in the 2NF
relation.

• Identify functional dependencies in the
relation.

• If transitive dependencies exist on the
primary key remove them by placing them
in a new relation along with copy of their
determinant.
Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
3NF Conversion Results

• Prevent referential integrity violation by
PROJECT (PROJ_NUM, PROJ_NAME)
ASSIGN (PROJ_NUM, EMP_NUM, HOURS)
EMPLOYEE (EMP_NUM, EMP_NAME, JOB_CLASS)
JOB (JOB_CODE, JOB_DESCRIPTION, CHG_HOUR)

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
General Definitions of 2NF and
3NF
• Second normal form (2NF)
– A relation that is in 1NF and every non-
primary-key attribute is fully functionally
dependent on any candidate key.

• Third normal form (3NF)
– A relation that is in 1NF and 2NF and in
which no non-primary-key attribute is
transitively dependent on any candidate key.

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Boyce–Codd Normal Form
(BCNF)
• Based on functional dependencies that
take into account all candidate keys in a
relation, however BCNF also has
general definition of 3NF.

• BCNF - A relation is in BCNF if and only
if every determinant is a candidate key.

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Boyce–Codd normal form
(BCNF)
• Difference between 3NF and BCNF is that for
a functional dependency A  B, 3NF allows
this dependency in a relation if B is a primary-
key attribute and A is not a candidate key.
• Whereas, BCNF insists that for this
dependency to remain in a relation, A must be
a candidate key.
• Every relation in BCNF is also in 3NF.
However, relation in 3NF may not be in BCNF.

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Boyce–Codd normal form
(BCNF)
• Violation of BCNF is quite rare.

• Potential to violate BCNF may occur in a
relation that:
– contains two (or more) composite candidate
keys;
– the candidate keys overlap (i.e. have at least
one attribute in common).

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
3NF Table Not in BCNF

Figure 4.7

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Decomposition of Table
Structure to Meet BCNF

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
BCNF Conversion Results

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Review of Normalization (UNF
to BCNF)

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Review of Normalization (UNF
to BCNF)

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Review of Normalization (UNF
to BCNF)

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Review of Normalization (UNF
to BCNF)

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Fourth Normal Form (4NF)
• Although BCNF removes anomalies due to
functional dependencies, another type of
dependency called a multi-valued dependency
(MVD) can also cause data redundancy.

• Possible existence of MVDs in a relation is due
to 1NF and can result in data redundancy.

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Fourth Normal Form (4NF) -
MVD
• Dependency between attributes (for
example, A, B, and C) in a relation, such
that for each value of A there is a set of
values for B and a set of values for C.
However, set of values for B and C are
independent of each other.

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Fourth Normal Form (4NF)
• MVD between attributes A, B, and C in a
relation using the following notation:
A  B
A  C

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Fourth Normal Form (4NF)
• MVD can be further defined as being trivial or
nontrivial.
– MVD A  B in relation R is defined as
being trivial if (a) B is a subset of A or (b) A 
B = R.
– MVD is defined as being nontrivial if neither
(a) nor (b) are satisfied.
– Trivial MVD does not specify a constraint on a
relation, while a nontrivial MVD does specify a
constraint.

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Fourth Normal Form (4NF)
• Defined as a relation that is in BCNF and
contains no nontrivial MVDs.

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
4NF - Example

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
3NF Table Not in BCNF

Figure 4.7

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Decomposition of Table
Structure to Meet BCNF

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Decomposition into BCNF

Figure 4.9
Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
4NF Conversion Results

Set of Tables in 4NF

Multivalued Dependencies (an employee can work for many services and
on many projects
Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Denormalization
• Normalization is one of many database design
goals
• Normalized table requirements
– Loss of system speed
• Normalization purity is difficult to sustain due to
conflict in:
– Design efficiency
– Information requirements
– Processing

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Unnormalized Table Defects
• Indexing more cumbersome
• No simple strategies for creating views

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU
Summary
• We will use normalization in database
design to create a set of relations in 3FN
normal form:
– Each entity has a unique primary key, and each
attribute depends upon the primary key
– No partial dependency
– No transitive dependency

Deepak Gour, Faculty – DBMS,
School of Engineering, SPSU

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