Question

Consider the universal relation \(R=\\{A, B, C, D, E, F, G, H, I, J\\}\) and the set of func- \\[ \text { tional dependencies } F=\\{\\{A, B\\} \rightarrow\\{C\\},\\{A\\} \rightarrow\\{D, E\\},\\{B\\} \rightarrow\\{F\\},\\{F\\} \rightarrow\\{G, H\\},\\{D\\} \rightarrow \\] \(\\{I, J\\} .\) What is the key for \(R ?\) Decompose \(R\) into \(2 \mathrm{NF}\) and then \(3 \mathrm{NF}\) relations.

Short answer

Key for R is {A, B}. It is decomposed into 2NF as R1(A, D, E), R2(A, B, C), R3(B, F), R4(F, G, H), R5(D, I, J) and these are also in 3NF.

Step-by-step solution

Find Closure

With the given functional dependencies, find the closure of each attribute sets. For instance, the closure of {A} can be found as {A, D, E, I, J} because A directly determines D and E, and D can further determine I and J.

Determine Key

A key is a set of one or more columns that can identify a record uniquely in a relational database. Looking at the closure, {A, B} can determine all other attributes in the relation. So, {A, B} is the key for the relation R.

Decompose into 2NF

A relation is in 2NF if and only if, for every non-prime attribute that is not itself part of a candidate key, each of its dependencies are on the entire key. We can decompose R into 2NF as follows: R1(A, D, E), R2(A, B, C), R3(B, F), R4(F, G, H), R5(D, I, J). In all of these subsets, the left side of each functional dependency is a key for the relation.

Decompose into 3NF

A relation is in 3NF if and only if it is in 2NF and the non-key attribute is fully functionally dependent on the primary key. Our 2NF relations obtained in previous step are already in 3NF.

Key concepts

relational database

A relational database is a structured way of storing data using tables, also known as relations. Each table is made up of rows and columns, where rows represent records and columns signify attributes. In a relational database, we can use SQL (Structured Query Language) to access and manage the data.

Key characteristics of relational databases include:

• **Rigidity and Structure**: Data is organized strictly in tables, adhering to specified schemas.
• **Data Integrity**: Supports constraints to ensure accuracy and consistency of data.
• **Relationships**: Links between tables are created using foreign keys, which reference columns in other tables.
• **Normalization**: The process of organizing data to reduce redundancy and improve data integrity.

Each relation or table can be a part of a larger set referred to as a universal relation. For instance, in our problem, the relation R includes several attributes such as \(A, B, C, D, E, F, G, H, I, J\). In the context of a relational database, we consider how these attributes interact and adhere to certain rules, known as functional dependencies. These dependencies are crucial because they help define relationships and ensure the efficiency and reliability of the database.

third normal form (3NF)

Third Normal Form (3NF) is a level of database normalization. It's aimed at reducing redundancy and ensuring that data dependencies are logical. For a table to comply with 3NF, it must first be in Second Normal Form (2NF).

Each criterion for 3NF can be broken down as follows:

• The table must be in 2NF: This means it should not have partial dependencies, where non-key attributes are dependent on only a part of the candidate key.
• Every non-key attribute is directly dependent on the primary key and not on another non-key attribute. This is called transitive dependency.

To achieve 3NF, a table must eliminate these transitive dependencies by decomposing the table into smaller tables without losing any data integrity. In our exercise, once we reached 2NF, we noticed no such transitive dependencies are present, so our tables are already in 3NF.

3NF is significant because it ensures that each piece of data is stored only once, reducing storage costs and making it easier to maintain the database. Any updates, deletions, or insertions will be made with consistency across the entire database system.

candidate key

A candidate key is a crucial concept in the study of relational databases. It refers to a set of one or more columns that can uniquely identify each row in a table. These columns are unique across the database; no two rows have the same values in all columns of a candidate key.

Characteristics of candidate keys include:

• **Uniqueness**: A candidate key's values must be unique for each row.
• **Minimalism**: The key should have the smallest number of columns necessary to maintain uniqueness.
• **Non-redundancy**: Removing any attribute from the candidate key would mean it no longer has the uniqueness property.

In our exercise, we're dealing with a universal relation \(R\) and identifying what can uniquely determine all other attributes. By examining functional dependencies and their closures, {A, B} is found to be a candidate key because it can uniquely identify the combination of all other attributes in \(R\).

Candidate keys are fundamental because they ensure every record can be retrieved and distinguished from others. By having efficient candidate keys, databases enhance performance for data retrieval, consistency, and integrity across operations.