Week 12 Discussion
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Closure of a Set of Functional Dependencies
In the context of relational databases, the closure of a set, often referred to as the
"functional dependency closure" or simply "closure," is a fundamental concept that plays a
crucial role in database design and normalization. The closure of a set of functional dependencies
represents the comprehensive collection of all functional dependencies that can be logically
deduced or implied from the given set. It serves as a powerful tool for understanding the
underlying relationships between attributes within a database schema, enabling database
designers to ensure data integrity, reduce redundancy, and optimize the structure of their
relational databases. By systematically deriving all possible dependencies, the closure provides a
clear and formalized way to uncover the intrinsic relationships that govern the organization of
data in a relational database.
The following example is from an article by Nisha (n.d.). Consider a relation R with
attributes A, B, C, D, E, F, and G, and the following set of functional dependencies: A → BC, BC
→ DE, D → F, and CF → G. We can find the closure of various attributes and attribute sets
within this relational schema.
Closure of attribute A:
A+ = {A} (trivial)
A+ = {A, B, C} (using A → BC)
A+ = {A, B, C, D, E} (using BC → DE)
A+ = {A, B, C, D, E, F} (using D → F)
A+ = {A, B, C, D, E, F, G} (using CF → G)
Thus, the closure of attribute A is A+ = {A, B, C, D, E, F, G}.
Closure of attribute D:
D+ = {D} (trivial)
D+ = {D, F} (using D → F)
We cannot determine any other attributes using attributes D and F. Thus, D+ = {D, F}.
Closure of attribute set {B, C}:
{B, C}+ = {B, C} (trivial)
{B, C}+ = {B, C, D, E} (using BC → DE)
{B, C}+ = {B, C, D, E, F} (using D → F)
{B, C}+ = {B, C, D, E, F, G} (using CF → G)
Hence, the closure of the attribute set {B, C} is {B, C, D, E, F, G}.
References
Elmasri, R., & Navathe, S. B. (2015). Fundamentals of Database Systems (7th ed.). Pearson
Education (US). https://reader2.yuzu.com/books/9780133971224
Nisha, C. D. (n.d.). DBMS : Closure properties - little flower college.
https://littleflowercollege.edu.in/upload/e_contents/files/187d8c7967172ef5cc75d9b2a70
9d6cf.pdf
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