DATABASE THOERY

1. DATABASE THOERY
FACILITATOR ALEX

2. INTRODUCTION
Database. A shared collection of logically related data,
and a description of this data, designed to meet the
information needs of an organization.
A DBMS is a software system that enables users to
define, create, maintain, and control access to the
database.
 A relation is a table with columns and rows.
An attribute is a named column of a relation.
Relation schema is a named relation defined by a set of

3. CONT’’
A domain is the set of allowable values for one or more
attributes.
A tuple is a row/record of a relation.
Degree of a relation is the number of attributes it
contains.
Cardinality of a relation is the number of tuples it
contains.
Relational database is a collection of normalized
relations with distinct relation names.

4. CONT’’
Relation =Table= File.
Tuple=Row=Record.
Attribute=Column=Field

5. PROPERTIES OF RELATIONS/TABLE
Must have a distinct name from all other relation
names.
Each cell of the relation contains exactly one atomic
(single) value.
Each attribute has a distinct name.
Each tuple is distinct; there are no duplicate tuples.
The values of an attribute are all from the same
domain

6. KEYS
A superkey is a set of one or more attributes that, taken
collectively, allow us to identify uniquely a tuple in the
relation.
Example
In a relation/table instructor{ID,NAME,DEPARTMENT
NAME}
{ID}
{ID,NAME}
{ID,DEPARTMENT NAME}
{NAME,DEPARTMENT NAME}

7. CONT’’
The ID attribute of the relation INSTRUCTOR is sufficient to
distinguish one instructor tuple from 29 another.Thus, ID is
a super key.
 The name attribute of instructor, on the other hand, is not a
super key, because several instructors might have the same
name.
And a combination of name and dept name is sufficient to
distinguish among members of the instructor relation.
 ID and name together can distinguish instructor tuples,
their combination, {ID, name}

8. CANDIDATE KEY
Candidate A superkey such that no proper subset is a
superkey within the key relation.
INSTRUCTOR TABLE {ID} is the candidate key and
{NAME,DEPARTMENT NAME}.
Properties
Irreducibility-in each tuple of R, the values of K uniquely
identify that tuple.
Uniqueness-no proper subset of K has the uniqueness
property.

9. PRIMARY KEY
PRIMARY KEY. The candidate key that is selected to identify tuples
uniquely within the key relation.
IN THE RELATION INSTRUCTOR
The primary key is {ID}
Properties
The value of primary key can never be NULL.
 The value of primary key must always be unique.
 The values of primary key can never be changed i.e. no updation is
possible.
 The value of primary key must be assigned when inserting a record.
 A relation is allowed to have only one primary key.

10. FOREIGN KEY
An attribute ‘X’ is called as a
foreign key to some other
attribute ‘Y’ when its values are
dependent on the values of
attribute ‘Y’.

12. FUNCTIONAL DEPENDENCY
Functional dependency (FD) is a set of constraints between two
attributes in a relation. Functional dependency says that if two tuples
have same values for attributes A1, A2,..., An, then those two tuples must
have to have same values for attributes B1, B2, ..., Bn.
 Functional dependency is represented by an arrow sign ( ) that is,
→
X Y, where X functionally determines Y.The left-hand side attributes
→
determine the values of attributes on the right-hand side.
Armstrong's Axioms
 If F is a set of functional dependencies then the closure of F, denoted as
F+, is the set of all functional dependencies logically implied by F.
Armstrong's Axioms are a set of rules, that when applied repeatedly,
generates a closure of functional dependencies.

13. CONT’’
Reflexive rule If alpha is a set of attributes and beta
−
is_subset_of alpha, then alpha holds beta.
Augmentation rule If a b holds and y is attribute
− →
set, then ay by also holds.That is adding attributes
→
in dependencies, does not change the basic
dependencies.
Transitivity rule Same as transitive rule in algebra,
−
if a b holds and b c holds, then a c also holds. a
→ → →
b is called as a functionally that determines b.
→

14. NORMALIZATION
Normalization is a formal method that can be used to identify relations
based on their keys and the functional dependencies among their
attributes.
It is a method of reducing duplication/data redundancy of a
relation/table.
Normal forms
First Normal A relation in which the intersection of each row and column
contains Form (1NF) one and only one value.
Second Normal A relation that is in First Normal Form and every non-
primary-key Form (2NF) attribute is fully functionally dependent on the
primary key.
Third Normal A relation that is in First and Second Normal Form and in
which no Form (3NF) non-primary-key attribute is transitively dependent

15. CONT’’
Trivial Functional Dependency
Trivial If a functional dependency (FD) X Y holds, where Y is
− →
a subset of X, then it is called a trivial FD.Trivial FDs always hold.
Non-trivial If an FD X Y holds, where Y is not a subset of X,
− →
then it is called a non-trivial FD.
Completely non-trivial If an FD X Y holds, where x intersect Y
− →
= , it is said to be a completely non-trivial FD.
Φ