Question

Consider the following two sets of functional dependencies: $$\begin{gathered} F= \\ A\to C,AC\to \text{)}\text{(}D,E\to AD,E\to H \end{gathered}$$ and $$\begin{gathered} G= \\ A\to CD,E\to AH \\ . \end{gathered}$$ Check whether they are equivalent.

Short answer

The equivalence of two sets of functional dependencies F and G is determined by the comparison of their closures, which are computed with the aid of Armstrong's axioms. The sets are considered equivalent if the closures infer each other.

Step-by-step solution

Finding the Closure for Set 1 (F)

Apply the Armstrong's axioms (reflexivity, augmentation, and transitivity) to obtain the closure of the functional dependency set F. Start with the given dependencies and move around the attributes utilising these axioms. An attribute A transitively depends on attribute B if there are attributes X and Y such that A depends on X and X depends on Y, then A depends on Y. Remember that the closure is the complete set of all the dependencies that can be derived from F using Armstrong's axioms.

Finding the Closure for Set 2 (G)

Follow the same process for G as was done for F. Apply Armstrong's axioms to obtain the closure of this functional dependency set G. The closure is the complete set of all the dependencies that can be derived from G using Armstrong's axioms.

Comparing the Closures

After computing the closure for F and G, compare the two closures. If the closures infer each other, i.e., every dependency in the closure of F is there in the closure of G, and vice versa, the two sets of functional dependencies are equivalent. If not, they are not equivalent.

Key concepts

Database Systems

When we talk about Database Systems, we refer to complex software designed to manage, store, retrieve, and manipulate data. These systems are essential for modern businesses, government agencies, educational institutions, and practically any organization that deals with a substantial amount of information.

Central to database systems is the concept of data modeling, which includes defining how data elements interrelate and how they are processed and stored. An important part of data modeling is ensuring that the database maintains integrity, avoids redundancy, and is efficient in operation. Functional dependencies are one of the tools used to achieve these goals, helping define relationships between data elements and, as a result, establishing the structure of database tables.

Armstrong's Axioms

To deeply understand how different elements in a database relate to one another, it's crucial to get to grips with Armstrong's Axioms. These are a set of rules used to infer all the possible functional dependencies from a given set of dependencies. Knowing these axioms by heart is essential, as they form the basis for reasoning about functional dependencies.

Reflexivity

If attribute B is a subset of attribute A, then A determines B. This seems intuitive - if you have the whole, you surely have the part.

Augmentation

If A determines B, then adding extra attributes to A will still mean that A determines B. Essentially, you're just bringing more to the table without changing the existing relationships.

Transitivity

If A determines B, and B determines C, then A determines C. It's like a chain reaction – one link leads to the next.

Functional Dependency Closure

The term Functional Dependency Closure, usually represented as F⁺, is often mentioned in the context of database systems. This is essentially a set containing all the functional dependencies that are implied by an original set of functional dependencies. In simpler words, it's the 'complete package' of rules about how attributes can determine one another in a database schema.

To find the closure of a set of functional dependencies, you methodically apply Armstrong's axioms until no more dependencies can be derived. This process aids in database normalization, the practice of organizing database components to reduce redundancy and improve data integrity. The closure helps you see not just the explicit dependencies but also those that can be deduced from them, ensuring a thorough understanding of the database's structure.

Attribute Dependency

Understanding Attribute Dependency is fundamental in grasping how data is related within a database. This concept refers to the way in which the value of one attribute (or a group of attributes) is related to the value of another attribute (or group of attributes) within a database. For example, in a University database, a student's ID may uniquely determine their name, major, and email address.

Knowing which attributes are dependent on others is crucial for database design and is often exploited to ensure data integrity and optimization. By carefully planning attribute dependencies, database designers can avoid unnecessary data replication and make the system more efficient, especially when it comes to retrieval and update operations.