The document outlines rules of inference for functional dependencies in a database management system, including reflexivity, augmentation, transitive dependency, and others. It discusses the identification of candidate keys using the closure method through various examples and attribute sets. The document also addresses the mapping from conceptual schema to logical schema.

1. Database Management System
By Assist Professor Malik M Ali

2. Rules of Inference for Functional Dependencies
Let A,B,C and D be subsets of attributes of a Relation R
Reflexivity
If A  B where B C A. This also implies that A  A
always holds. FDs of this type is known as trivial FDs
 A  B then A  A U B
A  B then A  A n B
Augmentation
If A  B, then A,C  B,C
Transitive dependency
A  B, B  C then A  C

3. Rules of Inference for Functional Dependencies
Additivity or Union
If A  B and A  C, then A  B,C
Decomposition
If A  B,C then A  B and A  C
PseudoTransitivity
If A  B and B,C  D then A,C  D

4. 1. Relation R{A,B,C,D}
FD {A  B, B  C, C  D}
A+ = ABCD
B+ = BCD
C+ = CD
Only One Candidate Key Identified i.e. A
2. Relation R{A,B,C,D}
 FD {A  B, B  C, C  D, D  A}
 A+ = ABCD
 B+ = BCDA
 C+ = CDAB
 D+ = DABC
 FOUR CANDIDATE KEYS IDENTIFIED
Closure Method

5. Relation R as Eight Attributes  R{A,B,C,D,E,F,G,H}
F = {CH  G, A  BC, B  CFH, EA, FEG}
Identify number of Candidate Keys in R?
D+ = D
1. AD+ = ADBCFHEG
2. ED+ = EDABCFHG
3. FD+ = DEGABCFH
4. BD+ = CDFHEGAB
Closure Method

6. Closure Method
Relation R as five Attributes  R{A,B,C,D,E}
FD = {A  B, BC  D, EC, DA}
Identify number of Candidate Keys in R?
AE = ABCED
BE = ?
CE = ?

7. Closure of Attributes
Is A a key for R?
Compute Closure of A and if it equals all attributes of that
relation than A is a key.
NIC_Num  StdName, StdAddress,
Percentage
HighSchoolCode  BoardofEdu,
HighschoolName, HighschoolCity
Percentage  Priority
{NIC_Num, HSCode}+
{NIC_Num, HSCode, StdName, StdAddress, Percentage,
Priority, BoardofEdu, HighschoolName, HighschoolCity}

8. Mapping of Conceptual Schema to Logical
Schema