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.

Practice

1. Which of the following is NOT one of Armstrong's axioms used in database theory?

easy

A. Reflexivity

B. Augmentation

C. Transitivity

D. Normalization

Solution

1. Step 1: Recall Armstrong's axioms

   Armstrong's axioms include Reflexivity, Augmentation, and Transitivity.

2. Step 2: Identify the option not in the axioms

   Normalization is a database design process, not an axiom.

3. Final Answer:

   Normalization -> Option D

4. Quick Check:

   Armstrong's axioms exclude Normalization [OK]

Hint: Remember the three axioms: Reflexivity, Augmentation, Transitivity [OK]

Common Mistakes:

• Confusing normalization with axioms
• Thinking all database terms are axioms
• Mixing up augmentation with normalization

2. Which Armstrong's axiom states that if Y is a subset of X, then X functionally determines Y?

easy

A. Reflexivity

B. Transitivity

C. Augmentation

D. Decomposition

Solution

1. Step 1: Understand the axiom definitions

   Reflexivity states if Y is subset of X, then X determines Y.

2. Step 2: Match the statement to the axiom

   The given statement matches Reflexivity exactly.

3. Final Answer:

   Reflexivity -> Option A

4. Quick Check:

   Subset implies functional dependency by Reflexivity [OK]

Hint: Subset implies dependency = Reflexivity [OK]

Common Mistakes:

• Confusing reflexivity with transitivity
• Choosing decomposition which is not an axiom
• Mixing augmentation with reflexivity

3. Given the functional dependencies: A -> B and B -> C, what can we infer using Armstrong's axioms?

medium

A. A -> C

B. B -> A

C. C -> A

D. C -> B

Solution

1. Step 1: Identify the given dependencies

   We have A determines B, and B determines C.

2. Step 2: Apply Transitivity axiom

   Transitivity says if A -> B and B -> C, then A -> C.

3. Final Answer:

   A -> C -> Option A

4. Quick Check:

   Transitivity gives A -> C [OK]

Hint: Chain dependencies to find indirect ones using Transitivity [OK]

Common Mistakes:

• Reversing dependencies incorrectly
• Ignoring transitivity rule
• Assuming dependencies go backward

4. Identify the error in this application of Armstrong's axioms: Given A -> B, the conclusion is B -> A.

medium

A. Correct use of augmentation

B. Proper use of reflexivity

C. Mistakenly reversed dependency

D. Valid transitivity application

Solution

1. Step 1: Analyze the given dependency

   A -> B means A determines B, not the other way around.

2. Step 2: Check the conclusion

   Concluding B -> A reverses the dependency, which Armstrong's axioms do not allow.

3. Final Answer:

   Mistakenly reversed dependency -> Option C

4. Quick Check:

   Dependencies do not reverse automatically [OK]

Hint: Dependencies don't reverse unless explicitly stated [OK]

Common Mistakes:

• Assuming dependencies are symmetric
• Confusing augmentation with reversal
• Ignoring direction of functional dependency

5. If you have A -> B and A -> C, which Armstrong's axiom allows you to combine these into A -> BC?

hard

A. Transitivity

B. Union

C. Decomposition

D. Augmentation

Solution

1. Step 1: Understand the given dependencies

   A determines B and A determines C separately.

2. Step 2: Apply the Union axiom

   Union allows combining A -> B and A -> C into A -> BC.

3. Final Answer:

   Union -> Option B

4. Quick Check:

   Union combines dependencies into one [OK]

Hint: Combine dependencies with same left side using Union [OK]

Common Mistakes:

• Confusing union with augmentation
• Using decomposition instead of union
• Applying transitivity incorrectly here