Canonical cover in DBMS Theory - Practice Problems & Coding Challenges

Challenge - 5 Problems

🧠 Conceptual

intermediate

What is the main purpose of a canonical cover in database design?

Choose the best description of the main purpose of a canonical cover for a set of functional dependencies.

A To list all possible functional dependencies in a relation

B To maximize the number of functional dependencies for better normalization

C To remove all keys from the set of functional dependencies

D To find a minimal set of functional dependencies that is equivalent to the original set

💡 Hint

Think about simplifying the set of dependencies without changing what they represent.

A canonical cover is a minimal set of functional dependencies that preserves the original dependencies' meaning. It removes redundancy and simplifies analysis.

📋 Factual

intermediate

Which step is NOT part of finding a canonical cover?

Select the step that is not involved in computing a canonical cover.

A Add new functional dependencies to increase coverage

B Remove extraneous attributes from the left side of dependencies

C Remove redundant functional dependencies

D Decompose functional dependencies with multiple attributes on the right side into single attributes

💡 Hint

Consider whether adding dependencies helps simplify or complicate the set.

Finding a canonical cover involves simplifying and reducing dependencies, not adding new ones.

🔍 Analysis

advanced

What is the canonical cover of the following set of functional dependencies?

F = { A -> BC, B -> C, A -> B, AB -> C }

Identify the canonical cover of the given functional dependencies.

A { A -> B, B -> C }

B { A -> BC, B -> C }

C { A -> B, AB -> C }

D { A -> C, B -> C }

💡 Hint

Try to remove redundant dependencies and extraneous attributes step by step.

Decompose A -> BC into A -> B and A -> C. Since B -> C exists, A -> C is redundant. Also, AB -> C is redundant because A -> B and B -> C imply AB -> C. So the minimal set is { A -> B, B -> C }.

❓ Reasoning

advanced

Why is it important to remove extraneous attributes from the left side of a functional dependency when computing a canonical cover?

Choose the best explanation for removing extraneous attributes from the left side of dependencies.

A To make the left side contain all attributes of the relation

B To increase the number of dependencies for better normalization

C To ensure the dependency is minimal and does not contain unnecessary attributes

D To convert the dependency into a multivalued dependency

💡 Hint

Think about what 'extraneous' means in this context.

Removing extraneous attributes makes the dependency minimal, which helps simplify the set and avoid redundancy.

❓ Comparison

expert

Compare the sets of functional dependencies before and after computing the canonical cover.

Given F = { A -> BC, B -> C, A -> B, AB -> C }, which statement is true?

Choose the correct statement about the relationship between the original set and its canonical cover.

A The canonical cover has more dependencies and a larger closure than the original set

B The canonical cover has fewer dependencies but represents the same closure as the original set

C The canonical cover removes some dependencies and changes the closure

D The canonical cover is identical to the original set

💡 Hint

Recall what equivalence means in terms of functional dependencies.

A canonical cover is a minimal equivalent set of dependencies that has the same closure as the original set but fewer dependencies.

Practice

1. What is the main purpose of a canonical cover in database design?

easy

A. To simplify and minimize the set of functional dependencies

B. To increase the number of functional dependencies

C. To add redundant dependencies for safety

D. To create more complex database schemas

Solution

1. Step 1: Understand the definition of canonical cover

   A canonical cover is a minimal set of functional dependencies that preserves the original dependencies without redundancy.

2. Step 2: Identify the purpose in database design

   It simplifies the design by removing redundant dependencies and splitting dependencies with multiple attributes on the right side.

3. Final Answer:

   To simplify and minimize the set of functional dependencies -> Option A

4. Quick Check:

   Canonical cover = minimal functional dependencies [OK]

Hint: Canonical cover means minimal and simple dependencies [OK]

Common Mistakes:

• Thinking it adds more dependencies
• Confusing it with normalization
• Assuming it complicates the schema

2. Which of the following is the correct step when creating a canonical cover?

easy

A. Remove extraneous attributes from dependencies

B. Add extra attributes to the left side

C. Combine all dependencies into one

D. Ignore dependencies with single attributes

Solution

1. Step 1: Recall the process of forming a canonical cover

   One key step is to remove extraneous attributes from the left or right side of functional dependencies to simplify them.

2. Step 2: Evaluate the options

   Removing extraneous attributes is correct, while adding extra attributes or ignoring dependencies is incorrect.

3. Final Answer:

   Remove extraneous attributes from dependencies -> Option A

4. Quick Check:

   Remove extraneous attributes = canonical cover step [OK]

Hint: Remove unnecessary parts to simplify dependencies [OK]

Common Mistakes:

• Adding attributes instead of removing
• Merging all dependencies incorrectly
• Ignoring single attribute dependencies

3. Given the functional dependencies:
F = { A -> BC, B -> C, A -> B }
What is the canonical cover of F?

medium

A. { A -> B, A -> C }

B. { A -> BC, B -> C }

C. { A -> B, B -> C }

D. { A -> B, B -> C, A -> C }

Solution

1. Step 1: Split dependencies with multiple attributes on right

   Split A -> BC into A -> B and A -> C.

2. Step 2: Remove redundant dependencies

   Since B -> C exists, and A -> B exists, A -> C is implied and redundant.

3. Final Answer:

   { A -> B, B -> C } -> Option C

4. Quick Check:

   Canonical cover removes redundant dependencies [OK]

Hint: Split right side, then remove implied dependencies [OK]

Common Mistakes:

• Not splitting multi-attribute dependencies
• Keeping redundant dependencies
• Confusing implied dependencies

4. Identify the error in this canonical cover step:
F = { A -> BC, B -> C }
Step: Remove C from A -> BC because B -> C exists.
Is this step correct?

medium

A. No, because C is not extraneous in A -> BC

B. Yes, because C is extraneous in A -> BC

C. Yes, because B -> C covers C

D. No, because B -> C is unrelated

Solution

1. Step 1: Check if C is extraneous in A -> BC

   Since B -> C exists, C can be derived from B, so C is extraneous on the right side of A -> BC.

2. Step 2: Confirm correctness of removal

   Removing C from A -> BC to get A -> B is valid and simplifies the dependencies.

3. Final Answer:

   Yes, because C is extraneous in A -> BC -> Option B

4. Quick Check:

   Extraneous attribute removal is valid [OK]

Hint: If attribute can be derived, it's extraneous [OK]

Common Mistakes:

• Thinking all right-side attributes are essential
• Ignoring dependency chains
• Confusing extraneous with essential attributes

5. You have the dependencies:
F = { AB -> C, A -> B, B -> C }
What is the canonical cover of F?

hard

A. { A -> B, AB -> C }

B. { AB -> C, A -> B }

C. { A -> B, B -> C, AB -> C }

D. { A -> B, B -> C }

Solution

1. Step 1: Check for extraneous attributes on left side

   In AB -> C, check if A or B is extraneous. Since B -> C exists, A is extraneous.

2. Step 2: Simplify AB -> C to B -> C and remove redundant dependencies

   Now F becomes { A -> B, B -> C }. This set is minimal and covers all dependencies.

3. Final Answer:

   { A -> B, B -> C } -> Option D

4. Quick Check:

   Remove extraneous left attributes and redundant dependencies [OK]

Hint: Remove extraneous left attributes to simplify [OK]

Common Mistakes:

• Not checking left side for extraneous attributes
• Keeping redundant dependencies
• Assuming all attributes on left are needed