Question

Prove or disprove the following inference rules for functional dependencies. A proof can be made either by a proof argument or by using inference rules IR1 through IR3. A disproof should be performed by demonstrating a relation instance that satisfies the conditions and functional dependencies in the left-hand side of the inference rule but does not satisfy the dependencies in the right-hand side. a. \(\\{W \rightarrow Y, X \rightarrow Z\\} \vDash\\{W X \rightarrow Y\\}\) b. \(\\{X \rightarrow Y\\}\) and \(Y \supseteq Z \vDash\\{X \rightarrow Z\\}\) ?. \(\\{X \rightarrow Y, X \rightarrow W, W Y \rightarrow Z\\} \vDash\\{X \rightarrow Z\\}\) d. \(\\{X Y \rightarrow Z, Y \rightarrow W\\} \vDash\\{X W \rightarrow Z\\}\) e. \(\\{X \rightarrow Z, Y \rightarrow Z\\} \vDash\\{X \rightarrow Y\\}\) f. \(\quad\\{X \rightarrow Y, X Y \rightarrow Z\\} \vDash\\{X \rightarrow Z\\}\) \(\mathrm{g} .\\{X \rightarrow Y, Z \rightarrow W\\} \vDash\\{X Z \rightarrow Y W\\}\) h. \(\\{X Y \rightarrow Z, Z \rightarrow X\\} \vDash\\{Z \rightarrow Y\\}\) ¡. \(\\{X \rightarrow Y, Y \rightarrow Z\\} \vDash\\{X \rightarrow Y Z\\}\) j. \(\quad\\{X Y \rightarrow Z, Z \rightarrow W\\} \vDash\\{X \rightarrow W\\}\)

Short answer

The rules that can be inferred are rules a, b, c, f, and i. The remaining rules can't be inferred due to the lack of a proper knowledge of dependencies.

Step-by-step solution

Exercise a

From the dependency set, apply the augmentation rule on \(W \rightarrow Y\) and \(X \rightarrow Z\) to get \(WX \rightarrow YX\) and \(WX \rightarrow ZX\) respectively. By applying the union rule, you get \(WX \rightarrow YZX\) and thus \(WX \rightarrow Y\).

Exercise b

Since \(Y \supseteq Z\), you know that \(Y \rightarrow Z\). By applying the transitivity rule on \(X \rightarrow Y\) and \(Y \rightarrow Z\), it infers that \(X \rightarrow Z\).

Exercise c

By applying the augmentation rule on \(X \rightarrow Y\) and \(X \rightarrow W\), you get \(X \rightarrow YW\). Now apply the transitivity rule with \(XYW \rightarrow Z\), it infers that \(X \rightarrow Z\).

Exercise d

You can't infer \(XW \rightarrow Z\) as there is no relation between X and W.

Exercise e

You can't infer \(X \rightarrow Y\) because there isn't any dependency of X on Y.

Exercise f

By using the augmentation rule and making \(X \rightarrow YX\). Now applying the transitivity rule with \(XY \rightarrow Z\), we infer \(X \rightarrow Z\).

Exercise g

You can't infer \(XZ \rightarrow YW\) as W is completely independent of the other attributes.

Exercise h

You can't infer \(Z \rightarrow Y\) as there is no direct functional dependency of Z on Y.

Exercise i

By applying the transitivity rule on \(X \rightarrow Y\) and \(Y \rightarrow Z\), it infers that \(X \rightarrow Z\), and thus \(X \rightarrow YZ\).

Exercise j

You can't infer \(X \rightarrow W\) as there isn't any dependency between X and W.

Key concepts

Inference Rules

Inference rules in the context of functional dependencies are logical constructs that allow us to derive new functional dependencies from a given set. When dealing with databases, these rules help maintain the integrity of the data by enabling us to deduce if certain dependencies hold based on the ones already known. Some of the most fundamental inference rules include:

• Reflexivity: If a set of attributes X contains an attribute A, then X can determine A. Formally, if A is a subset of X, then X → A.
• Augmentation: If X determines Y, then X combined with any other attribute Z will determine Y combined with Z. This is usually denoted by saying that if X → Y, then XZ → YZ.
• Transitivity: If X determines Y and Y determines Z, then X will also determine Z. This is a common rule used to draw chains of dependencies.

Understanding and applying these rules is crucial for database theory as they ensure functional dependencies are logically consistent and aid in database normalization.

Proof and Disproof Techniques

In database theory, proving or disproving an inference rule requires demonstrating that a dependency logically follows (or does not follow) from a given set of dependencies. Proof can be achieved by constructing an argument that shows the logical sequence of dependencies using inference rules, while disproof involves finding a counterexample.
To prove a rule, follow these steps:

• Identify Known Dependencies: Start with the functional dependencies given in the problem.
• Apply Inference Rules: Use reflexivity, augmentation, transitivity, or other relevant rules to derive the desired dependency.
• Logical Sequence: Ensure each step logically follows from the previous ones to reach the conclusion.

For disproof, attempt to:

• Construct a Relation Instance: Create a table that conforms to the given dependencies but fails the one being tested.
• Check for Contradictions: Ensure that while satisfying the left-hand dependencies, the right-hand dependencies do not hold.

By carefully applying these techniques, one can accurately determine the validity of the dependencies in question.

Augmentation Rule

The augmentation rule is a powerful tool in working with functional dependencies. This rule states that if you have a functional dependency X → Y, then by adding any extra set of attributes to both sides, the dependency still holds. Mathematically, if you have X → Y, you can infer XZ → YZ for any attribute set Z.
This rule is useful when you need to build larger dependencies from smaller ones. For example, if you know that a key attribute set X determines a non-key attribute Y, then any combination that includes X will also determine a combination that includes Y. This is particularly handy when designing a database structure or when you need to decompose relations while preserving dependencies.
To apply the augmentation rule, simply:

• Add the same attributes to both sides of a dependency (if needed).
• Ensure that the modified dependency remains logically consistent within the context of the database schema.
• Use these modified dependencies to help analyze or organize data within the database.

Transitivity Rule

The transitivity rule is a fundamental part of reasoning in database theory. This rule helps to infer indirect functional dependencies. If you have the dependencies X → Y and Y → Z, you can deduce that X → Z. This mirrors the transitive property found in mathematics.
The transitivity rule is particularly useful when you're dealing with chains of dependencies. In practice, if a certain attribute depends on another through a series of dependencies, transitivity allows you to make these indirect relationships explicit.
Here's how you might typically use the transitivity rule:

• Identify a "chain" of dependencies from the known set. For instance, if attribute X determines Y and Y determines Z.
• Apply the transitivity to conclude that X must determine Z.
• Use the derived dependency to simplify or optimize a database structure.

Understanding transitivity can greatly aid in database design and ensuring data integrity by enabling more profound insights and connections between data sets.

Database Theory

Database theory provides the foundational concepts and logical frameworks for understanding the complexities of data management and organization. At its core, database theory encompasses the principles of how databases should be structured, accessed, and maintained.
Key components of database theory include:

• Functional Dependencies: These indicate relationships between attributes, serving as the basis for analyzing and structuring data.
• Normalization: A process aimed at organizing data to minimize redundancy and dependency. Functional dependencies play an essential role here.
• Database Design: The art of constructing a database structure that is efficient, reliable, and scalable, often leveraging concepts like ER modeling and schema refinement.

Understanding and applying database theory is crucial for anyone dealing with relational databases, as it ensures that data is both logically consistent and practically accessible. It helps in creating databases that are not only efficient in terms of storage but also are capable of ensuring data integrity and reducing anomalies.