Armstrong's axioms in DBMS Theory - Practice Problems & Coding Challenges

Challenge - 5 Problems

🧠 Conceptual

intermediate

Understanding Armstrong's Axioms: Reflexivity

Which of the following best describes the Reflexivity axiom in Armstrong's axioms for functional dependencies?

A If X functionally determines Y and Y functionally determines Z, then X functionally determines Z.

B If X functionally determines Y, then Y functionally determines X.

C If X functionally determines Y, then X combined with any attribute Z functionally determines Y.

D If Y is a subset of X, then X functionally determines Y (X → Y).

💡 Hint

Think about when a set of attributes determines itself or its parts.

The Reflexivity axiom states that if Y is a subset of X, then X determines Y. This means any set of attributes determines its own subsets.

🧠 Conceptual

intermediate

Applying Armstrong's Axioms: Augmentation

Which statement correctly explains the Augmentation axiom?

A If X functionally determines Y and Y functionally determines Z, then X functionally determines Z.

B If X functionally determines Y, then Y functionally determines X.

C If X functionally determines Y, then adding Z to the left side preserves the dependency: XZ → Y.

D If Y is a subset of X, then X functionally determines Y.

💡 Hint

Consider what happens when you add extra attributes to both sides of a dependency.

Augmentation means if X → Y, then adding attributes Z to the left side keeps the dependency valid: XZ → Y.

❓ Reasoning

advanced

Using Armstrong's Axioms: Transitivity Application

Given the functional dependencies: A → B and B → C, what can be concluded using Armstrong's axioms?

A C → A

B A → C

C B → A

D C → B

💡 Hint

Think about chaining dependencies to find indirect relationships.

Transitivity states that if A → B and B → C, then A → C.

🔍 Analysis

advanced

Identifying Invalid Functional Dependency

Which of the following functional dependencies violates Armstrong's axioms?

A If X → Y, then Y → X.

B If X → Y and Y → Z, then X → Z.

C If Y is subset of X, then X → Y.

D If X → Y, then XZ → YZ.

💡 Hint

Recall which axioms allow reversing dependencies.

Armstrong's axioms do not allow reversing dependencies. If X → Y, it does not imply Y → X.

❓ Comparison

expert

Comparing Armstrong's Axioms and Derived Rules

Which of the following is a derived rule that can be proven using Armstrong's axioms but is not one of the original axioms?

A Union: If X → Y and X → Z, then X → YZ.

B Transitivity: If X → Y and Y → Z, then X → Z.

C Augmentation: If X → Y, then XZ → YZ.

D Reflexivity: If Y is subset of X, then X → Y.

💡 Hint

Think about which rules are basic axioms and which are proven from them.

Union is a derived rule proven from Armstrong's axioms, not one of the original three axioms.