Relational Database Design Functional Dependency – definition, trivial and non-trivial FD, Armstrong's Axioms/Inference Rules

Database Management
System
Bhoomika Patel , Assistant Professor
Computer Science & Engineering

Relational Databse Design
CHAPTER-5

Content
s
● Functional Dependency – definition, trivial and non-trivial FD
● Armstrong's Axioms/Inference Rules
● Closure of FD
● Closure of Attributes
● Candidate Key
● Finding a Candidate Key
● Decomposition (Lossy & Lossless)
● Database Anomalies
● normalization – 1Nf, 2NF, 3NF, BCNF, 4NF, 5NF

● Functional Dependency:
● Brief overview of databases and the importance of data integrity.
● Introduction to Functional Dependencies (FD) as a concept ensuring data consistency.
● Functional Dependency (FD) is a constraint between two sets of attributes in a relation from a
database.
● Notation: X→YX rightarrow YX→Y
● Meaning: Attribute Y is functionally dependent on attribute X if each value of X is associated with
precisely one value of Y.
● Example:
● Table with attributes: StudentID, StudentName, CourseID, CourseName
● Example FD: StudentID→StudentNameStudentID rightarrow
StudentNameStudentID→StudentName, CourseID→CourseNameCourseID rightarrow
CourseNameCourseID→CourseName
● Explanation of the relationship between the attributes.

● Trivial Functional Dependency:
● Definition: A functional dependency X→YX rightarrow YX→Y is trivial if YYY is a subset of XXX.
● Example: If X={A,B}X = {A, B}X={A,B}, then A→AA rightarrow AA→A and {A,B}→A{A, B}
rightarrow A{A,B}→A are trivial FDs.
● Explanation: These dependencies hold true by default.

● Non-Trivial Functional Dependency:
● Definition: A functional dependency X→YX rightarrow YX→Y is non-trivial if YYY is not a subset of
XXX.
● Example: If X={A}X = {A}X={A} and Y={B}Y = {B}Y={B}, then A→BA rightarrow BA→B is non-
trivial.
● Explanation: These dependencies provide meaningful constraints in database design.

Relational Database Design Functional Dependency – definition, trivial and non-trivial FD, Armstrong's Axioms/Inference Rules

● Candidatekey:
● A candidate key is a minimal set of attributes that can uniquely identify a tuple in a relation.
● Every relation can have one or more candidate keys.
● Finding candidate key:
Given a relation and a set of functional dependencies:
● Relation R(A, B, C, D)
● FDs: {A → B, C → D}
Step-by-step process to identify candidate keys.

● Decomposition:
● Decomposition is the process of breaking down a relation into two or more
sub-relations.
● Purpose: To eliminate redundancy, prevent anomalies, and ensure data
integrity.
● Types:
● Lossy Decomposition
● Lossless Decomposition

● Lossy Decomposition:
● Definition: Decomposition where the original relation cannot be perfectly
reconstructed from the decomposed sub-relations.
● Example: Illustrative example showing data loss after decomposition.
● Consequences: Data anomalies, potential loss of information.
● Given relation R(A, B, C).
● Decomposed into R1(A, B) and R2(B, C).
● Loss of information due to redundancy in B.
● Illustration of how original relation cannot be accurately reconstructed.

● Lossless Decomposition:
● Definition: Decomposition where the original relation can be perfectly
reconstructed from the decomposed sub-relations.
● Conditions: Dependency preservation and ensuring common attributes form a
super key.
● Importance: Ensures data integrity and prevents anomalies.
● Given relation R(A, B, C).
● Decomposed into R1(A, B) and R2(A, C).
● Reconstruction: Natural join of R1 and R2 restores the original relation.
● Illustration of how the original relation is accurately reconstructed.

● Database Anomalies:
● Definition: Inconsistencies or errors in a database due to improper database
design.
● Causes: Typically caused by redundancy and lack of normalization.

First Normal Form (1NF)
○ Definition: A relation is in 1NF if it contains only atomic (indivisible) values.
○ Rules: No repeating groups, no multi-valued attributes.
○ Example: Converting a table with repeating groups to 1NF.
Example: First Normal Form (1NF)
○ Initial table with repeating groups.
○ Conversion to 1NF by eliminating repeating groups and ensuring atomic values.

Second Normal Form (2NF)
○ Definition: A relation is in 2NF if it is in 1NF and all non-key attributes are fully dependent on
the primary key.
○ Rule: Eliminate partial dependencies.
○ Example: Converting a table from 1NF to 2NF.
example: Second Normal Form (2NF)
○ Initial table in 1NF with partial dependencies.
○ Conversion to 2NF by removing partial dependencies and creating separate tables.

Third Normal Form (3NF):
○ Definition: A relation is in 3NF if it is in 2NF and all the attributes are functionally
dependent only on the primary key.
○ Rule: Eliminate transitive dependencies.
Example: Third Normal Form (3NF)
○ Initial table in 2NF with transitive dependencies.
○ Conversion to 3NF by removing transitive dependencies and creating separate
tables.

Boyce-Codd Normal Form (BCNF)
○ Definition: A relation is in BCNF if it is in 3NF and every determinant is a candidate key.
○ Rule: Handle situations where 3NF is insufficient.
○ Example: Converting a table from 3NF to BCNF.
Example: Boyce-Codd Normal Form (BCNF)
○ Initial table in 3NF with anomalies.
○ Conversion to BCNF by ensuring every determinant is a candidate key.

Fourth Normal Form (4NF)
○ Definition: A relation is in 4NF if it is in BCNF and has no multi-valued dependencies.
○ Rule: Eliminate multi-valued dependencies.
○ Example: Converting a table from BCNF to 4NF.
Example: Fourth Normal Form (4NF)
○ Initial table in BCNF with multi-valued dependencies.
○ Conversion to 4NF by removing multi-valued dependencies and creating separate tables.

Fifth Normal Form (5NF)
○ Definition: A relation is in 5NF if it is in 4NF and has no join dependencies.
○ Rule: Eliminate join dependencies.
Example: Fifth Normal Form (5NF)
○ Initial table in 4NF with join dependencies.
○ Conversion to 5NF by removing join dependencies and ensuring that the table can be
decomposed into smaller tables without loss of information.

Relational Database Design Functional Dependency – definition, trivial and non-trivial FD, Armstrong's Axioms/Inference Rules

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