chap 10 dbms.pptx

Chapter 10
Functional Dependencies and Normalization
for Relational Databases

Chapter Outline
1 Informal Design Guidelines for Relational Databases
1.1Semantics of the Relation Attributes
1.2 Redundant Information in tuples & Update Anomalies
1.3 Null Values in Tuples
1.4 Spurious Tuples
2 Functional Dependencies (FDs)
2.1 Definition of FD
2.2 Inference Rules for FDs
2.3 Equivalence of Sets of FDs
2.4 Minimal Sets of FDs

Chapter Outline(contd.)
3 Normal Forms Based on Primary Keys
3.1 Normalization of Relations
3.2 Practical Use of Normal Forms
3.3 Definitions of Keys and Attributes
Participating in Keys
3.4 First Normal Form
3.5 Second Normal Form
3.6 Third Normal Form
4 General Normal Form Definitions (For Multiple Keys)
5 BCNF (Boyce-Codd Normal Form)

1 Informal Design Guidelines for Relational
Databases (1)
• What is relational database design?
The grouping of attributes to form "good" relation schemas
• Two levels of relation schemas
– The logical "user view" level
– The storage "base relation" level
• Design is concerned mainly with base relations
• What are the criteria for "good" base relations?

Informal Design Guidelines for Relational
Databases (2)
• We first discuss informal guidelines for good
relational design
• Then we discuss formal concepts of functional
dependencies and normal forms
- 1NF (First Normal Form)
- 2NF (Second Normal Form)
- 3NF (Third Normal Form)
- BCNF (Boyce-Codd Normal Form)
• Additional types of dependencies, further normal
forms, relational design algorithms by synthesis are
discussed in Chapter 11

1.1 Semantics of the Relation Attributes
GUIDELINE 1: Informally, each tuple in a relation
should represent one entity or relationship
instance. (Applies to individual relations and their
attributes).
• Attributes of different entities (EMPLOYEEs, DEPARTMENTs, PROJECTs)
should not be mixed in the same relation
• Only foreign keys should be used to refer to other entities
• Entity and relationship attributes should be kept apart as much as
possible.
Bottom Line: Design a schema that can be explained
easily relation by relation. The semantics of
attributes should be easy to interpret.

Chapter 10-7
Figure 10.1 A simplified COMPANY
relational database schema
Note: The above figure is now called Figure 10.1 in Edition 4

1.2 Redundant Information in Tuples and Update
Anomalies
• Mixing attributes of multiple entities may cause
problems
• Information is stored redundantly wasting storage
• Problems with update anomalies
– Insertion anomalies
– Deletion anomalies
– Modification anomalies

EXAMPLE OF AN UPDATE ANOMALY (1)
Consider the relation:
EMP_PROJ ( Emp#, Proj#, Ename, Pname, No_hours)
• Update Anomaly: Changing the name of project
number P1 from “Billing” to “Customer-Accounting”
may cause this update to be made for all 100
employees working on project P1.

EXAMPLE OF AN UPDATE ANOMALY (2)
• Insert Anomaly: Cannot insert a project unless an
employee is assigned to .
Inversely - Cannot insert an employee unless an
he/she is assigned to a project.
• Delete Anomaly: When a project is deleted, it will
result in deleting all the employees who work on that
project. Alternately, if an employee is the sole
employee on a project, deleting that employee
would result in deleting the corresponding project.

Two relation schemas suffering from update
anomalies

Example States for EMP_DEPT and EMP_PROJ

Guideline to Redundant Information in Tuples
and Update Anomalies
• GUIDELINE 2: Design a schema that does not suffer
from the insertion, deletion and update anomalies. If
there are any present, then note them so that
applications can be made to take them into account

1.3 Null Values in Tuples
GUIDELINE 3: Relations should be designed such that
their tuples will have as few NULL values as possible
• Attributes that are NULL frequently could be placed
in separate relations (with the primary key)
• Reasons for nulls:
– attribute not applicable or invalid
– attribute value unknown (may exist)
– value known to exist, but unavailable

1.4 Spurious Tuples
• Spurious Tuple: A record or row in the database
formed because of linking the two tables incorrectly.
These additional tuples may not be required to
present the information in the database. The user
may need to normalize the table to eliminate these
tuples.
• Bad designs for a relational database may result in
erroneous results for certain JOIN operations
• The "lossless join" property is used to guarantee
meaningful results for join operations
GUIDELINE 4: The relations should be designed to
satisfy the lossless join condition. No spurious tuples
should be generated by doing a natural-join of any
relations.

Spurious Tuples (2)
There are two important properties of
decompositions:
(a) non-additive or losslessness of the corresponding
join
(b) preservation of the functional dependencies.
Note that property (a) is extremely important and
cannot be sacrificed. Property (b) is less stringent
and may be sacrificed. (See Chapter 11).

Module 6 17 7/10/2023
Example of Spurious Tuples

2.1 Functional Dependencies (1)
• Functional dependencies (FDs) are used to specify
formal measures of the "goodness" of relational
designs
• FDs and keys are used to define normal forms for
relations
• FDs are constraints that are derived from the
meaning and interrelationships of the data
attributes
• A set of attributes X functionally determines a set
of attributes Y if the value of X determines a
unique value for Y

Module 6 19 7/10/2023
Generation of spurious tuples
• The two relations EMP_PROJ1 and EMP_LOCS as the base
relations of EMP_PROJ, is not a good schema design.
• Problem is if a Natural Join is performed on the above two
relations it produces more tuples than original set of tuples
in EMP_PROJ.
• These additional tuples that were not in EMP_PROJ are
called spurious tuples because they represent spurious or
wrong information that is not valid.
• This is because the PLOCATION attribute which is used for
joining is neither a primary key, nor a foreign key in either
EMP_LOCS AND EMP_PROJ1.

Module 6 20 7/10/2023
Example of Spurious Tuples contd

Module 6 21 7/10/2023
4. Spurious Tuples
• Bad designs for a relational database may result in
erroneous results for certain JOIN operations
• The "lossless join" property is used to guarantee
meaningful results for join operations
• GUIDELINE 4:
– Design relation schemas so that they can be joined
with equality conditions on attributes that are either
primary keys or foreign keys in a way that guarantees
that no spurious tuples are generated.

Module 6 22 7/10/2023
Spurious Tuples
• There are two important properties of decompositions:
– Non-additive or losslessness of the corresponding join
– Preservation of the functional dependencies.
• Note that:
– Property (a) is extremely important and cannot be sacrificed.
– Property (b) is less stringent and may be sacrificed.

Functional Dependencies (2)
• X -> Y holds if whenever two tuples have the same value for
X, they must have the same value for Y
• For any two tuples t1 and t2 in any relation instance r(R): If
t1[X]=t2[X], then t1[Y]=t2[Y]
• X -> Y in R specifies a constraint on all relation instances r(R)
• Written as X -> Y; can be displayed graphically on a relation
schema as in Figures. ( denoted by the arrow: ).
• FDs are derived from the real-world constraints on the
attributes

Examples of FD constraints (1)
• social security number determines employee name
SSN -> ENAME
• project number determines project name and
location
PNUMBER -> {PNAME, PLOCATION}
• employee ssn and project number determines the
hours per week that the employee works on the
project
{SSN, PNUMBER} -> HOURS

Examples of FD constraints (2)
• An FD is a property of the attributes in the schema R
• The constraint must hold on every relation instance
r(R)
• If K is a key of R, then K functionally determines all
attributes in R (since we never have two distinct
tuples with t1[K]=t2[K])

2.2 Inference Rules for FDs (1)
• Given a set of FDs F, we can infer additional FDs that
hold whenever the FDs in F hold
Armstrong's inference rules:
IR1. (Reflexive) If Y subset-of X, then X -> Y
IR2. (Augmentation) If X -> Y, then XZ -> YZ
(Notation: XZ stands for X U Z)
IR3. (Transitive) If X -> Y and Y -> Z, then X -> Z
• IR1, IR2, IR3 form a sound and complete set of
inference rules

Inference Rules for FDs (2)
Some additional inference rules that are useful:
(Decomposition) If X -> YZ, then X -> Y and X -> Z
(Union) If X -> Y and X -> Z, then X -> YZ
(Psuedotransitivity) If X -> Y and WY -> Z, then WX -> Z
• The last three inference rules, as well as any other
inference rules, can be deduced from IR1, IR2, and IR3
(completeness property)

Inference Rules for FDs (3)
• Closure of a set F of FDs is the set F+ of all FDs that can
be inferred from F
• Closure of a set of attributes X with respect to F is the set
X + of all attributes that are functionally determined by X
• X + can be calculated by repeatedly applying IR1, IR2, IR3
using the FDs in F

2.3 Equivalence of Sets of FDs
• Two sets of FDs F and G are equivalent if:
- every FD in F can be inferred from G, and
- every FD in G can be inferred from F
• Hence, F and G are equivalent if F + =G +
Definition: F covers G if every FD in G can be inferred from F
(i.e., if G + subset-of F +)
• F and G are equivalent if F covers G and G covers F
• There is an algorithm for checking equivalence of sets of
FDs

2.4 Minimal Sets of FDs (1)
• A set of FDs is minimal if it satisfies the following
conditions:
(1) Every dependency in F has a single attribute for its RHS.
(2) We cannot remove any dependency from F and have a set of
dependencies that is equivalent to F.
(3) We cannot replace any dependency X -> A in F with a dependency
Y -> A, where Y proper-subset-of X ( Y subset-of X) and still have
a set of dependencies that is equivalent to F.

Minimal Sets of FDs (2)
• Every set of FDs has an equivalent minimal set
• There can be several equivalent minimal sets
• There is no simple algorithm for computing a minimal set
of FDs that is equivalent to a set F of FDs
• To synthesize a set of relations, we assume that we start
with a set of dependencies that is a minimal set (e.g., see
algorithms 11.2 and 11.4)

3 Normal Forms Based on Primary Keys
3.1 Normalization of Relations
Participating in Keys
3.5 Second Normal Form
3.6 Third Normal Form

3.1 Normalization of Relations (1)
• Normalization: The process of decomposing
unsatisfactory "bad" relations by breaking up their
attributes into smaller relations
• Normal form: Condition using keys and FDs of a
relation to certify whether a relation schema is in a
particular normal form

Normalization of Relations (2)
• 2NF, 3NF, BCNF based on keys and FDs of a relation
schema
• 4NF based on keys, multi-valued dependencies : MVDs;
5NF based on keys, join dependencies : JDs (Chapter 11)
• Additional properties may be needed to ensure a good
relational design (lossless join, dependency preservation;
Chapter 11)

• Normalization is carried out in practice so that the resulting designs
are of high quality and meet the desirable properties
• The practical utility of these normal forms becomes questionable
when the constraints on which they are based are hard to
understand or to detect
• The database designers need not normalize to the highest possible
normal form. (usually up to 3NF, BCNF or 4NF)
• Denormalization: the process of storing the join of higher normal
form relations as a base relation—which is in a lower normal form

Participating in Keys (1)
• A superkey of a relation schema R = {A1, A2, ...., An} is a
set of attributes S subset-of R with the property that no
two tuples t1 and t2 in any legal relation state r of R will
have t1[S] = t2[S]
• A key K is a superkey with the additional property that
removal of any attribute from K will cause K not to be a
superkey any more.

Definitions of Keys and Attributes Participating in
Keys (2)
• If a relation schema has more than one key, each is called
a candidate key. One of the candidate keys is arbitrarily
designated to be the primary key, and the others are
called secondary keys.
• A Prime attribute must be a member of some candidate
key
• A Nonprime attribute is not a prime attribute—that is, it
is not a member of any candidate key.

• Disallows composite attributes, multivalued
attributes, and nested relations; attributes
whose values for an individual tuple are non-
atomic
• Considered to be part of the definition of
relation

Figure 10.8 Normalization into 1NF

Figure 10.9 Normalization nested relations into
1NF

3.3 Second Normal Form (1)
• Uses the concepts of FDs, primary key
Definitions:
• Prime attribute - attribute that is member of the
primary key K
• Full functional dependency - a FD Y -> Z where
removal of any attribute from Y means the FD does
not hold any more
Examples: - {SSN, PNUMBER} -> HOURS is a full FD since neither
SSN -> HOURS nor PNUMBER -> HOURS hold
- {SSN, PNUMBER} -> ENAME is not a full FD (it is called a partial
dependency ) since SSN -> ENAME also holds

Second Normal Form (2)
• A relation schema R is in second normal form
(2NF) if every non-prime attribute A in R is fully
functionally dependent on the primary key
• R can be decomposed into 2NF relations via the
process of 2NF normalization

Figure 10.10 Normalizing into 2NF and 3NF

Figure 10.11 Normalization into 2NF and 3NF

3.4 Third Normal Form (1)
Definition:
• Transitive functional dependency - a FD X -> Z that can be
derived from two FDs X -> Y and Y -> Z
Examples:
- SSN -> DMGRSSN is a transitive FD since
SSN -> DNUMBER and DNUMBER -> DMGRSSN hold
- SSN -> ENAME is non-transitive since there is no set of attributes
X where SSN -> X and X -> ENAME

Third Normal Form (2)
• A relation schema R is in third normal form (3NF) if it is
in 2NF and no non-prime attribute A in R is transitively
dependent on the primary key
• R can be decomposed into 3NF relations via the process
of 3NF normalization
NOTE:
In X -> Y and Y -> Z, with X as the primary key, we consider this a problem only
if Y is not a candidate key. When Y is a candidate key, there is no problem with
the transitive dependency .
E.g., Consider EMP (SSN, Emp#, Salary ).
Here, SSN -> Emp# -> Salary and Emp# is a candidate key.

4 General Normal Form Definitions (For Multiple
Keys) (1)
• The above definitions consider the primary key only
• The following more general definitions take into
account relations with multiple candidate keys
• A relation schema R is in second normal form (2NF)
if every non-prime attribute A in R is fully
functionally dependent on every key of R

General Normal Form Definitions (2)
Definition:
• Superkey of relation schema R - a set of attributes S
of R that contains a key of R
• A relation schema R is in third normal form (3NF) if
whenever a FD X -> A holds in R, then either:
(a) X is a superkey of R, or
(b) A is a prime attribute of R
NOTE: Boyce-Codd normal form disallows condition (b) above

5 BCNF (Boyce-Codd Normal Form)
• A relation schema R is in Boyce-Codd Normal Form
(BCNF) if whenever an FD X -> A holds in R, then X is a
superkey of R
• Each normal form is strictly stronger than the previous one
– Every 2NF relation is in 1NF
– Every 3NF relation is in 2NF
– Every BCNF relation is in 3NF
• There exist relations that are in 3NF but not in BCNF
• The goal is to have each relation in BCNF (or 3NF)

Figure 10.12 Boyce-Codd normal form

Figure 10.13 a relation TEACH that is in 3NF but not in
BCNF

Achieving the BCNF by Decomposition (1)
• Two FDs exist in the relation TEACH:
fd1: { student, course} -> instructor
fd2: instructor -> course
• {student, course} is a candidate key for this relation and that the
dependencies shown follow the pattern in Figure 10.12 (b). So this
relation is in 3NF but not in BCNF
• A relation NOT in BCNF should be decomposed so as to meet this
property, while possibly forgoing the preservation of all functional
dependencies in the decomposed relations. (See Algorithm 11.3)

Achieving the BCNF by Decomposition (2)
• Three possible decompositions for relation TEACH
1. {student, instructor} and {student, course}
2. {course, instructor } and {course, student}
3. {instructor, course } and {instructor, student}
• All three decompositions will lose fd1. We have to settle for sacrificing the
functional dependency preservation. But we cannot sacrifice the non-additivity
property after decomposition.
• Out of the above three, only the 3rd decomposition will not generate spurious tuples
after join.(and hence has the non-additivity property).
• A test to determine whether a binary decomposition (decomposition into two
relations) is nonadditive (lossless) is discussed in section 11.1.4 under Property LJ1.
Verify that the third decomposition above meets the property.

• 22. Boyce-Codd Normal Form a. BCNF
• b. A R is in BCNF if whenever an FD X A holds in R, then X is a
super
• key of R
• c. Each normal form is strictly stronger than the previous one i.
Every 2NF relation is in 1NF
• ii. Every 3NF relation is in 2NF
• iii. Every BCNF relation is in 3NF
• d. There exist relations that are in 3NF but not in BCNF
• e. The goal is to have each relation in BCNF (or 3NF)
• f. LOTS1A relation

• i. Has primary key
• ii. All non-key columns are FD on entire
primary key iii. Now all table are stratified in 2
NF, 3 NF, BCNF

chap 10 dbms.pptx

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