Question

, Date_sold, Salesman#, Commission\%, Discount_amt) Assume that a car may be sold by multiple salesmen, and hence \(\\{\mathr… # Consider the following relation: CAR_SALE(Car#, Date_sold, Salesman#, Commission\%, Discount_amt) Assume that a car may be sold by multiple salesmen, and hence \)\\{\mathrm{CAR} \\#,\( SALESMAN is the primary key. Additional dependencies are Date_sold \)\rightarrow\( Discount amt and Salesman# \)\rightarrow\( Commission\% Based on the given primary key, is this relation in \)1 \mathrm{NF}, 2 \mathrm{NF}\(, or \)3 \mathrm{NF}$ ? Why or why not? How would you successively normalize it completely?

Short answer

The initial relation CAR_SALE was in 1NF but not in 2NF due to the presence of partial dependencies. After further normalization, the revised relations SALE, SALESMAN, and CAR_SALE are in 3NF.

Step-by-step solution

Verify 1NF

To verify if the relation, CAR_SALE, is in its 1NF, we need to check whether it has any repeating groups. The relation has no repeating groups because each record is unique with respect to the primary key which is the combination of CAR# and SALESMAN#. Therefore, the relation is in 1NF.

Verify 2NF

To verify if the relation is in 2NF, we must ensure that there is no partial dependency i.e., no non-prime attribute is dependent on any proper subset of any candidate key. In this case, the dependencies Date_sold → Discount_amt and Salesman# → Commission\% both suggest that there are attributes (Discount_amt and Commission\%) that are dependent on a proper subset of the candidate key (CAR#, SALESMAN#). So, the given relation is not in 2NF.

Normalize to 2NF

The solution to normalize the relation to 2NF is to remove the partial dependencies. We create two new relations: SALE(Date_sold, Discount_amt) and SALESMAN(Salesman#, Commission\%). Now, the CAR_SALE relation will be (Car#, Date_sold, Salesman#).

Verify 3NF and Normalize if Necessary

To verify if the updated relations are in 3NF, we must ensure there's no transitive dependency. The relations SALE, SALESMAN, and CAR_SALE satisfy the conditions for 3NF as they do not have any non-prime attribute that is dependent on any non-candidate key attribute. Therefore, these relations are in 3NF.

Key concepts

First Normal Form (1NF)

The First Normal Form, or 1NF, is the foundation of the normalization process in database management. It focuses on eliminating repeating groups within a table. To ensure that a table is in 1NF, each cell must contain only atomic values, meaning that no cell can hold multiple data items at once. This means no arrays or lists in a single cell.

To achieve 1NF, each column must store unique values that cannot be further divided. In the original problem, the CAR_SALE table has a composite primary key consisting of both Car# and Salesman#. As each entry is uniquely identified by this composite key with no repeating groups, the table is considered to already be in first normal form.

Second Normal Form (2NF)

Second Normal Form builds on the requirements of 1NF and requires the elimination of partial dependencies. A table is in 2NF if it is already in 1NF and all non-prime attributes are fully functionally dependent on the entire set of candidate keys.

In simpler terms, a non-prime attribute should not depend just on a part of a composite key. In the CAR_SALE relation, we encounter partial dependencies such as Date_sold → Discount_amt and Salesman# → Commission%. Here, Discount_amt and Commission% depend only on a portion of the composite key (Car#, Salesman#).

To normalize to 2NF, we must split the CAR_SALE table to eliminate these dependencies. This process involves creating new relations like SALE(Date_sold, Discount_amt) and SALESMAN(Salesman#, Commission%), ensuring that all non-prime attributes depend fully on the primary key.

Third Normal Form (3NF)

The Third Normal Form emphasizes removing transitive dependencies. A table is in 3NF if it is in 2NF and no non-prime attribute depends on another non-prime attribute. Essentially, non-prime attributes must directly relate to the candidate keys only.

After addressing the partial dependencies and crafting the SALE and SALESMAN tables, each table needs evaluation for transitive dependencies. In our revised setup, there are no additional attributes in any table that depend on other non-prime attributes, which means these tables already comply with the requirements for 3NF.

This normalization further ensures that data redundancy is minimized and integrity is enhanced, allowing for better consistency across database entries.

Functional Dependency

Functional Dependency in databases indicates a relationship between two attributes, typically between a primary key and non-prime attributes. If an attribute B is functionally dependent on attribute A, any value of A determines the value of B.

For the given problem, functional dependencies include Date_sold → Discount_amt and Salesman# → Commission%. These signify that each sale date is associated specifically with a discount amount, and each salesman with a commission percentage.

Understanding these relationships is crucial for breaking down tables non-redundantly in the normalization process, as they allow us to identify partial, full, or transitive dependencies.

Candidate Key

A Candidate Key is a minimal subset of attributes in a table that can uniquely identify a tuple. Each table must have at least one candidate key which serves as a possible choice for a primary key.

In the CAR_SALE relation from the exercise, the composite key (Car#, Salesman#) acts as the candidate key. This means that each combination of a car number and a salesman number uniquely identifies a sales record.

Discovering candidate keys is a foundational step in evaluating tables for normalization. Every non-prime attribute must be examined for functional dependencies in relation to these keys, to progress through 1NF, 2NF, and 3NF steps efficiently.