Concepts of Functional Dependency, rules and types. Also introduction to Boyce-Codd Normal Form (BCNF)

1. FUNCTIONAL DEPENDENCY
PRINCIPLES OF DATA AND KNOWLEDGE
BASE SYSTEMS
NAME: ALAA NOOR
MATRIKL NR.: 805642

2. What is Functional Dependency (FDs)?
Functional dependency is a relationship that exists when one attribute uniquely determines
another attribute.
FDs are a generalization of keys.
It is the central part of relational database design theory.
A functional dependency is denoted by an arrow "→". Such as the functional dependency of A on
B is represented by A →B

3. EXAMPLE OF FD:
Student_ID Name Age City
01 Alaa 21 Potsdam
02 Iftekhar 25 Potsdam
03 Aziha 22 Berlin
• Here if we know the value of
Student_ID, we can obtain the Name,
Age and City of that student. By this,
we can say that the City, Name, and
Age are functionally dependent on
Student_ID.
• Functional dependency can be written
as: Student_ID → Name, Age, City.
Table 1. Student table

4. Key Terms:
Key Terms Description
Axiom Axioms is a set of inference rules used to infer all
the functional dependencies on a relational
database.
Decomposition It is a rule that suggests if you have a table that
appears to contain two entities which are
determined by the same primary key(PK) then
you should consider breaking them up into two
different tables.
Dependent It is displayed on the right side of the functional
dependency.
Determinant It is displayed on the left side of the functional
dependency.
Union It suggests that if two tables are separate, and
the PK is the same, you should consider putting
them together.
Table 2. Key terms of Functional Dependency

5. Rules of Functional Dependency:
1. Reflexive rule : If B is a subset of A, then A→B holds by reflexivity rule.
2. Augmentation rule : It is also known as a partial dependency, it says if A->B
holds, then AC->BC for any attribute C also holds.
3. Transitivity rule : This rule is very much similar to the transitive rule in
algebra if A->B holds and B->C holds, then A->C also holds.

6. Types of Functional Dependency:
 Transitive Dependency
 Trivial functional dependency
 Non-trivial functional dependency
Figure 1. Types of FDs

7. Transitive Dependency:
A functional dependency is said to be transitive if it is indirectly formed by
two functional dependencies.
X -> Z is a transitive dependency if the following three functional
dependencies hold true:
• X->Y
• Y does not ->X
• Y->Z
A transitive dependency can only occur in a relation of three or more
attributes. This dependency helps us normalizing the database in 3NF (3rd
Normal Form).

8. Transitive Dependency Example:
Book Author Author Age
Game of Thrones George R. R. Martin 66
Harry Potter J.K. Rowling 49
Dying of the Light George R. R. Martin 66
• {Book} ->{Author} (if we know the book, we can know the author name)
• {Author} does not -> {Book}
• {Author} -> {Author_age}
Therefore according to the rule of transitive dependency:
• {Book} -> {Author_age} should hold, that makes sense because if we know the book
name we can know the author’s age.
Table 3. Book table

9. Trivial Functional Dependency:
The dependency of an attribute on a set of attributes is known as trivial
functional dependency if the set of attributes includes that attribute.
Symbolically:
• A ->B is trivial functional dependency if B is a subset of A.
• The following dependencies are also trivial: A->A & B->B

10. Trivial FD Example:
• {Company_Id, Company_Name} -> {Company_Id} is a trivial functional
dependency as Company_Id is a subset of {Company_Id, Company_Name}.
• Also, Company_Id -> Company_Id & Company_Name -> Company_Name are
trivial dependencies too.
Company_Id Company_Name
C2 Apple
C3 Facebook
C4 Google
Table 4. Company Table

11. Non-Trivial Functional Dependency
A nontrivial functional dependency occurs when A->B holds true where B is
not a subset of A.
For example:
Company_Id-> Company_Name
The above is non-trivial functional dependency since Company_Name is
not a subset of Company_Id.

12. Contd..
Any FD such as A1, . . . , An → B1, . . . , Bm will cause redundant storage unless A1,
. . . , An is a key, so that each combination of attribute values for A1, . . . , An can
occur only once.
Avoid (proper) FDs by transforming them into key constraints. This is what
normalization does.
Therefore BCNF is used here.

13. Boyce-Codd Normal Form (BCNF)
A relational schema R is considered to be in Boyce–Codd normal form (BCNF) if,
for every one of its dependencies A → B, one of the following conditions holds
true:
• A → B is a trivial functional dependency
• A relation R is in BCNF if and only if: whenever there is a nontrivial FD
A1, . . . , An → B1, . . . , Bm for R, it is the case that {A1, A2, ... , An} is a superkey
for R.
#That is, the left side of every nontrivial FD must be a superkey.

14. Example (Not in BCNF):
The nontrivial functional dependencies in the table are:
• author → nationality
• book title → genre, number of pages
We can easily see that the only KEY is the set {author, book title}
Author Nationality Book Title Genre Number of pages
William
Shakespeare
English The Tempest Comedy 215
Emily Brontë Irish Wuthering Heights Gothic 540
Jeffrey Ullman American Database Systems Textbook 500
Jennifer Widom American Database Systems Textbook 500
Table 5. Author table

15. Example (in BCNF):
Author Nationality
William Shakespeare English
Emily Brontë Irish
Jeffrey Ullman American
Jennifer Widom American
Author Book Title
William Shakespeare The Tempest
Emily Brontë Wuthering Heights
Jeffrey Ullman Database Systems
Jennifer Widom Database Systems
Table 6. Author1 table Table 7. Author2 table

16. Contd..
The functional dependencies for this schema are the same as before:
• author → nationality
• book title → genre, number of pages
The key of the first table is {author}. The key of the second table is {author, book title}. The key of the
third table is {book title}. There are no functional dependencies violating the BCNF rules, so the
schema is in Boyce-Codd Normal Form.
Book Title Genre Number of Pages
The Tempest Comedy 215
Wuthering Heights Gothic 540
Database Systems Textbook 500
Table 8. Author3 table

17. Advantages of Functional Dependency:
• Functional Dependency removes data redundancy where the same values
should not be repeated at multiple locations in the same database table.
• It maintains the quality of data in the database.
• FD allows clearly defined meanings and constraints of databases.
• It helps in identifying bad designs of the database.
• It expresses the facts about the database design.

18. References
H. Garcia-Molina, J. Ullman, and J.Widom. Database Systems: The Complete
Book. Prentice Hall Press, second edition, 2008. [Ch. 3]
https://www.guru99.com/dbms-functional-dependency.html
https://www.geeksforgeeks.org/types-of-functional-dependencies-in-dbms
https://www.tutorialspoint.com/Functional-dependency-in-DBMS
https://www.tutorialandexample.com/functional-dependencies/