# Common BCNF and 3NF Problems in Schema Design

In the realm of database design, normalization is a fundamental process that ensures data integrity, reduces redundancy, and enhances the efficiency of data management. This technique is crucial for anyone involved in database homework and plays a key role in creating schemas that are both reliable and scalable. Boyce-Codd Normal Form (BCNF) and Third Normal Form (3NF) are two critical levels of normalization that significantly impact how effectively a database operates. Understanding these forms is essential for tackling complex schema homework, as they address different aspects of functional dependency and schema design.

BCNF and 3NF provide structured approaches to eliminating redundancy and ensuring that each attribute is properly aligned with the schema’s keys. This blog aims to guide you through the process of identifying BCNF and 3NF violations, and performing the necessary decompositions to achieve optimal database schemas. We will cover practical examples and detailed steps to help you master these concepts effectively, paving the way for more efficient and reliable database management.

## Understanding BCNF and 3NF

Before diving into practical examples, it’s essential to understand the principles behind BCNF and 3NF:

**BCNF (Boyce-Codd Normal Form):**A relation schema is in BCNF if, for every functional dependency (FD) X → Y, X is a superkey. In other words, a relation is in BCNF if every determinant is a superkey. This form is a stricter version of 3NF and addresses some of its shortcomings.**3NF (Third Normal Form):**A relation schema is in 3NF if, for every FD X → Y, either X is a superkey, or Y is a prime attribute (an attribute that is part of some candidate key). 3NF aims to reduce redundancy while allowing some flexibility for FDs that do not violate the general integrity of the database.

## Analyzing BCNF Violations and Decomposition

Let’s explore how to identify BCNF violations and perform decompositions using various relation schemas and functional dependencies.

**1. Relation Schema:** R(A, B, C, D) with FDs ABC → D, CD → A

**Identifying BCNF Violations:
**

To determine if R(A, B, C, D) is in BCNF, we need to examine each FD and ensure that its left side is a superkey.

- FD ABC → D: ABC is a superkey. Thus, this FD does not violate BCNF.
- FD CD → A: CD is not a superkey (since it does not include all attributes needed to form a superkey). Therefore, this FD violates BCNF.

**Decomposition to Achieve BCNF:
**

To resolve the BCNF violation caused by CD → A, we decompose the relation R:

**1. Decompose based on CD → A:
**

- Create relation R1(C, D, A) to capture the FD CD → A.
- Create relation R2(A, B, C) to handle the remaining attributes and FDs.

**Each resulting relation is now in BCNF, as:
**

- In R1(C, D, A), CD is a superkey.
- In R2(A, B, C), ABC is a superkey.

**2. Relation Schema: **R(A, B, C, D) with FDs B → C and BD → A

**Identifying BCNF Violations:
**

- FD B → C: B is not a superkey, so this FD violates BCNF.
- FD BD → A: BD is a superkey. Therefore, this FD does not violate BCNF.

**Decomposition to Achieve BCNF:
**

To resolve the BCNF violation caused by B → C, we decompose the relation R:

**1. Decompose based on B → C:
**

- Create relation R1(B, C) to capture B → C.
- Create relation R2(A, B, D) to handle the remaining attributes and FDs.

**Both relations are now in BCNF:
**

- In R1(B, C), B is a superkey.
- In R2(A, B, D), BD is a superkey.

**3. Relation Schema: **R(A, B, C, D) with FDs ABC → D, BCD → A, CD → A, and ADB → C

**Identifying BCNF Violations:
**

**FD ABC → D:**ABC is a superkey.**FD BCD → A:**BCD is a superkey.**FD CD → A:**CD is not a superkey, so this FD violates BCNF.**FD ADB → C:**ADB is a superkey.

**Decomposition to Achieve BCNF:
**

To resolve the BCNF violation caused by CD → A, we decompose the relation R:

**1. Decompose based on CD → A:
**

- Create relation R1(C, D, A) to handle CD → A.
- Create relation R2(A, B, C) to capture the remaining attributes.

**Both relations are now in BCNF:
**

- In R1(C, D, A), CD is a superkey.
- In R2(A, B, C), ABC is a superkey.

**4. Relation Schema:** R(A, B, C, D) with FDs A → B, B → C, CD → A, and D → A

**Identifying BCNF Violations:
**

- FD A → B: A is not a superkey, so this FD violates BCNF.
- FD B → C: B is not a superkey, so this FD also violates BCNF.
- FD CD → A: CD is a superkey.
- FD D → A: D is not a superkey.

**Decomposition to Achieve BCNF:
**

To resolve the BCNF violations, we perform the following decompositions:

**1. Decompose based on A → B:
**

- Create relation R1(A, B) to capture A → B.
- Create relation R2(A, C, D) to handle the remaining attributes and FDs.

**2. Further decompose R2 based on D → A:
**

- Create relation R3(D, A) to handle D → A.
- Create relation R4(A, C) to capture the remaining attributes.

**All resulting relations are now in BCNF:
**

- R1(A, B) with A as a superkey.
- R3(D, A) with D as a superkey.
- R4(A, C) with A as a superkey.

**5. Relation Schema:** R(A, B, C, D, E) with FDs ABC → D, DE → C, and B → D

**Identifying BCNF Violations:
**

**FD ABC → D:**ABC is a superkey.**FD DE → C:**DE is not a superkey, so this FD violates BCNF.**FD B → D:**B is not a superkey, so this FD also violates BCNF.

**Decomposition to Achieve BCNF:
**

To resolve BCNF violations caused by DE → C and B → D, we decompose the relation R:

**1. Decompose based on DE → C:
**

- Create relation R1(D, E, C) to handle DE → C.
- Create relation R2(A, B, D, E) to manage the remaining attributes.

**2. Further decompose R2 based on B → D:
**

- Create relation R3(B, D) to handle B → D.
- Create relation R4(A, B, E) to capture the remaining attributes.

**All resulting relations are now in BCNF:
**

- R1(D, E, C) with DE as a superkey.
- R3(B, D) with B as a superkey.
- R4(A, B, E) with AB as a superkey.

## Exploring 3NF Violations and Decomposition

### Decomposing Relation Schema to 3NF

3NF focuses on eliminating transitive dependencies, ensuring that non-prime attributes are fully functionally dependent on a superkey. Let’s explore how to address 3NF violations:

**1. Relation Schema:** R(A, B, C, D) with FDs A → B and A → C

**Decomposition Based on A → B and A → C:
**

**1. Decompose into R1(A, B) and R2(A, C, D):
**

- R1(A, B) captures the dependency A → B.
- R2(A, C, D) manages the remaining attributes.

Both relations are in BCNF, hence in 3NF.

**2. Relation Schema: **R(A, B, C, D) with FDs A → B and B → C

**Decomposition Based on A → B:
**

**1. Decompose into R1(A, B) and R2(B, C, D):
**

- R1(A, B) captures A → B.
- R2(B, C, D) manages the remaining attributes.

Both relations are in BCNF, thus in 3NF.

**Alternative Decomposition Based on A → B and B → C:
**

**1. Decompose into R1(A, B, C) and R2(A, D):
**

- R1(A, B, C) handles A → B and B → C.
- R2(A, D) manages the remaining attributes.

Both decompositions result in relations that are in 3NF.

**3. Relation Schema: **R(A, B, C) with FD AB → C

**Decomposition to Ensure 3NF Compliance:
**

**1. Decompose into R1(A, B) and R2(B, C):
**

- R1(A, B) handles AB → C.
- 2(B, C) captures the remaining attributes.

**For instance, if R contains tuples (1, 2, 3) and (1, 2, 4):
**

- S will have (1, 2).
- T will have (2, 3) and (2, 4).
- The projections on S and T might not yield the original relation if tuples are inconsistent, indicating that while decomposition might help with normalization, it’s crucial to ensure that projections accurately reflect the original data.

## Conclusion

Database normalization is pivotal for creating well-structured, efficient, and reliable database schemas. By systematically identifying BCNF and 3NF violations and applying appropriate decompositions, you can enhance data integrity and reduce redundancy in your database design.

Mastering the art of normalization requires practice and a thorough understanding of functional dependencies and their implications on schema design. By following the outlined steps and examples, you can tackle complex normalization challenges and achieve optimal database schemas that support robust data management practices.