9.1 Functional dependency theory
3 min read•august 6, 2024
Functional dependencies are the backbone of database design, defining relationships between attributes. They're crucial for understanding how data is connected and organized within a database. Knowing these dependencies helps create efficient, normalized database structures.
and inference rules are your toolset for working with functional dependencies. These rules allow you to deduce new dependencies from existing ones, helping you uncover hidden relationships in your data and optimize your database schema.
Functional Dependencies
Key Concepts
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Top images from around the web for Key Concepts
Chapter 11 Functional Dependencies – Database Design – 2nd Edition View original
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Chapter 11 Functional Dependencies – Database Design – 2nd Edition View original
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- (FD) denotes a relationship between attributes where the value of one attribute determines the value of another
- Given relation R, attribute Y is functionally on attribute X () if each X value is associated with precisely one Y value
- refers to the attribute on the left-hand side of a functional dependency (X in X → Y)
- Dependent represents the attribute on the right-hand side of a functional dependency (Y in X → Y)
Types of Functional Dependencies
- occurs when an attribute depends on a composite key but not on any proper subset of that key
- If (X, Y) → Z, but neither X → Z nor Y → Z hold, then Z is fully functionally dependent on (X, Y)
- Partial functional dependency arises when an attribute depends on only part of a composite key
- If (X, Y) → Z and either X → Z or Y → Z hold, then Z is partially functionally dependent on (X, Y)
- Partial dependencies can lead to data redundancy and update anomalies in a relation
Attribute Closure and Inference Rules
Attribute Closure
- Closure of attributes refers to the set of all attributes functionally determined by a given set of attributes
- To find the closure of attribute set X (denoted as X+), start with X and repeatedly add attributes based on given functional dependencies until no more attributes can be added
- helps determine all functional dependencies that can be inferred from a given set of dependencies
Inference Rules
- Armstrong's axioms are a set of inference rules used to infer all functional dependencies on a relational schema
- : If Y ⊆ X, then X → Y
- : If X → Y, then XZ → YZ
- : If X → Y and Y → Z, then X → Z
- Additional inference rules derived from Armstrong's axioms include:
- : If X → Y and X → Z, then X → YZ
- : If X → YZ, then X → Y and X → Z
- : If X → Y and WY → Z, then WX → Z
Transitive Dependencies
Concept and Implications
- occurs when a non-prime attribute depends on another non-prime attribute
- If X → Y and Y → Z, but X is not functionally dependent on Z, then Z transitively depends on X via Y
- Transitive dependencies can lead to data redundancy and update anomalies in a relation
- Removing transitive dependencies is a key step in normalizing relations to higher normal forms (e.g., 3NF)
Example Scenario
- Consider a relation R(A, B, C) with functional dependencies A → B and B → C
- In this case, C transitively depends on A via B
- To eliminate the transitive dependency, decompose R into two relations:
- R1(A, B) with FD A → B
- R2(B, C) with FD B → C
- The resulting relations are free of transitive dependencies and in a higher normal form
Key Terms to Review (23)
Armstrong's Axioms: Armstrong's Axioms are a set of inference rules used in the field of database management to derive all functional dependencies in a relational database. These axioms provide a formal framework for reasoning about functional dependencies, which are critical for ensuring data integrity and normalization in database design. They consist of three primary rules: reflexivity, augmentation, and transitivity, which can be used to infer additional dependencies from existing ones.
Attribute closure: Attribute closure is a concept used in functional dependency theory that refers to the set of all attributes that can be functionally determined by a given set of attributes in a relational database. This concept helps in understanding the implications of functional dependencies and is essential for normalization, ensuring that all relevant data is included in a relation without redundancy.
Augmentation rule: The augmentation rule is a fundamental principle in functional dependency theory that states if a functional dependency holds, then adding attributes to both sides of the dependency will also hold. This means that if a set of attributes determines another set of attributes, then adding additional attributes to both the determinant and the dependent attributes preserves the dependency. It’s crucial for understanding how relationships between attributes behave when expanding the schema of a database.
Candidate key: A candidate key is a minimal set of attributes that uniquely identifies a record in a relational database table. Each candidate key can be used to uniquely retrieve data, ensuring that no two rows have the same value for this key, while also being the smallest possible set of attributes needed for that uniqueness. Understanding candidate keys is crucial as they directly relate to functional dependencies, normalization processes, and the establishment of relationships between tables.
Decomposition Rule: The decomposition rule is a fundamental principle in functional dependency theory that states if a relation has a functional dependency, it can be decomposed into smaller relations without losing any information. This rule helps to break down complex relations into simpler components, which makes it easier to manage and analyze the data while maintaining the integrity of the original relationships among the attributes.
Dependent: In the context of functional dependency theory, a dependent refers to an attribute or set of attributes in a relation that relies on another attribute or set of attributes for its value. This concept is crucial for understanding how data is structured and how relationships between different pieces of data can be defined, influencing normalization and database design.
Determinant: In database theory, a determinant is an attribute or a set of attributes that uniquely identifies a value in another attribute. It establishes a relationship between two sets of attributes, where the value of the determinant determines the value of the dependent attribute. Understanding determinants is crucial for recognizing functional dependencies, normalization, and ensuring data integrity in relational databases.
First normal form (1NF): First normal form (1NF) is a property of a relational database table that ensures the data is stored in a way that each column contains atomic values, and each entry in a column is of the same kind. This means that there are no repeating groups or arrays within a single column, allowing for clearer relationships and easier data manipulation. Achieving 1NF is a critical step in the normalization process and helps establish a solid foundation for further normalization, such as second and third normal forms.
Full functional dependency: Full functional dependency occurs when a non-key attribute is functionally dependent on the entire primary key, rather than just a part of it. This means that if you remove any attribute from the primary key, the non-key attribute would no longer be dependent on the remaining attributes. Understanding this concept is crucial for proper database design, as it helps ensure that all attributes are correctly associated with their keys, leading to more efficient data organization and retrieval.
Functional Dependency: Functional dependency is a relationship between attributes in a database, where one attribute uniquely determines another. This concept is crucial for understanding how data is organized and structured within a database, influencing the design of schemas and the normalization process to eliminate redundancy and ensure data integrity.
Insertion anomaly: An insertion anomaly occurs when certain attributes cannot be added to a database without the presence of other attributes. This usually happens in poorly designed databases, where data is not organized efficiently, leading to redundancy and complications when trying to insert new records. This concept highlights the importance of proper database design and normalization to avoid unnecessary dependencies and complexities.
Normalization: Normalization is a systematic approach to organizing data in a database to reduce redundancy and improve data integrity. It involves decomposing tables into smaller, related tables and defining relationships between them to ensure that each piece of data is stored only once, which minimizes the risk of anomalies during data operations.
Partial dependency: Partial dependency is a type of functional dependency that occurs in a relational database when a non-key attribute is functionally dependent on part of a composite primary key. This means that the value of the non-key attribute can be determined by only a portion of the primary key rather than the entire key, leading to potential redundancy and anomalies in the database.
Primary Key: A primary key is a unique identifier for each record in a database table, ensuring that no two rows can have the same key value. It plays a crucial role in maintaining the integrity of data by preventing duplicate entries and enabling efficient data retrieval.
Pseudotransitivity rule: The pseudotransitivity rule is a principle in functional dependency theory that deals with relationships among attributes in a relational database. It states that if a functional dependency X → Y holds and Y is also a determinant for another functional dependency Y → Z, then it can be inferred that X → Z holds under certain conditions. This rule helps in understanding how dependencies can be linked together to deduce new dependencies.
Reflexivity Rule: The reflexivity rule is a principle in functional dependency theory that states if a set of attributes contains another set of attributes, then the second set is functionally dependent on the first. This means that any attribute or combination of attributes in a relation can determine itself. This rule is foundational because it helps to establish the basic relationships within a database structure and contributes to understanding more complex dependencies.
Schema design: Schema design refers to the process of defining the structure, organization, and relationships of data in a database. It involves creating a blueprint that determines how data is stored, accessed, and managed, ensuring that it meets the needs of users while maintaining data integrity and efficiency. Proper schema design considers factors such as normalization, functional dependencies, and the use of different modeling techniques to represent entities and their relationships effectively.
Third Normal Form (3NF): Third Normal Form (3NF) is a database normalization condition that ensures all non-key attributes are functionally dependent only on the primary key and not on any other non-key attributes. This helps eliminate transitive dependencies, where a non-key attribute depends on another non-key attribute, thus reducing data redundancy and improving data integrity in relational databases.
Transitive dependency: Transitive dependency occurs when a non-key attribute depends on another non-key attribute, which in turn depends on a key attribute. This relationship can lead to redundancy and anomalies in data management. Understanding transitive dependencies is essential for structuring databases effectively, especially when establishing functional dependencies and ensuring that the database adheres to normal forms.
Transitivity rule: The transitivity rule is a fundamental principle in functional dependency theory that states if a functional dependency exists between a set of attributes, then certain implications can be inferred about other attributes. Specifically, if attribute A determines attribute B, and attribute B determines attribute C, then it logically follows that attribute A also determines attribute C. This rule helps in reasoning about relationships among attributes and simplifies the design of database schemas.
Union rule: The union rule is a principle in functional dependency theory that states if a set of attributes functionally determines another set of attributes, then the union of these sets also functionally determines any combination of their respective attributes. This principle is essential in understanding how attributes relate to each other and helps in analyzing and designing databases.
Update anomaly: An update anomaly occurs in a database when changes to a data item require multiple updates to ensure consistency, leading to potential data integrity issues. This problem often arises in poorly designed databases that do not conform to normalization principles, which help maintain data integrity through structured organization.
X → y: In the context of database systems, the notation 'x → y' represents a functional dependency where the attribute x uniquely determines the attribute y. This means that for each value of x, there is exactly one corresponding value of y. Understanding functional dependencies is essential for database normalization, ensuring data integrity and reducing redundancy.