Candidate key finding using closure in DBMS Theory - Practice Problems & Coding Challenges

Challenge - 5 Problems

🧠 Conceptual

intermediate

Understanding Closure in Candidate Key Identification

Given a set of attributes R = {A, B, C, D} and functional dependencies F = {A → B, B → C}, what is the closure of attribute A (denoted as A⁺)?

A {A, B}

B {A, B, C}

C {A, C}

D {A, B, C, D}

💡 Hint

Start with attribute A and add attributes that can be functionally determined step by step.

Starting with A, from A → B, add B. Then from B → C, add C. So A⁺ = {A, B, C}.

🔍 Analysis

intermediate

Determining Candidate Key Using Closure

Consider relation R = {A, B, C, D} with functional dependencies F = {A → B, B → C, C → D}. Which of the following is a candidate key?

A C

B D

C B

D A

💡 Hint

Find the closure of each attribute and check which covers all attributes in R.

Closure of A is {A, B, C, D} because A → B, B → C, and C → D. So A is a candidate key.

🧠 Conceptual

advanced

Effect of Adding Functional Dependencies on Candidate Keys

Given relation R = {A, B, C} and functional dependencies F = {A → B}, which of the following changes will make A no longer a candidate key?

A Add functional dependency C → A

B Add functional dependency B → A

C Add functional dependency B → C

D Add functional dependency C → B

💡 Hint

Consider how the closure of A changes with the new dependency.

If C → A is added, then A can be derived from C, so A is not minimal as a candidate key anymore.

❓ Reasoning

advanced

Minimality Check in Candidate Keys

Relation R = {A, B, C, D} has functional dependencies F = {A → B, B → C, A → D}. Which of the following sets is a candidate key?

A {B, D}

B {B, C}

C {A}

D {A, B}

💡 Hint

Check closure of each set and verify minimality.

Closure of {A} is {A, B, C, D} so it covers all attributes and is minimal. Other sets are either not minimal or do not cover all attributes.

❓ Comparison

expert

Comparing Candidate Keys Using Closure

Given relation R = {A, B, C, D, E} and functional dependencies F = {A → B, BC → D, D → E}, which of the following is a candidate key?

A {A, C}

B {A, B}

C {B, C}

D {A, B, C}

💡 Hint

Calculate closure of each set and check which covers all attributes.

Closure of {B, C} is {B, C, D, E} but missing A, so not candidate key. Closure of {A, C} is {A, B, C, D, E} covering all attributes and minimal. However, {A, C} is minimal and covers all attributes, so it is candidate key.

Practice

1. What does the closure of a set of attributes represent in candidate key finding?

easy

A. All attributes functionally determined by the set

B. The number of attributes in the set

C. The total number of functional dependencies

D. The primary key of the table

Solution

1. Step 1: Understand closure definition

   The closure of a set of attributes includes all attributes that can be functionally determined from that set using the given functional dependencies.

2. Step 2: Relate closure to candidate keys

   If the closure covers all attributes in the relation, the set is a superkey; minimal such sets are candidate keys.

3. Final Answer:

   All attributes functionally determined by the set -> Option A

4. Quick Check:

   Closure = attributes reachable from set [OK]

Hint: Closure shows all attributes a set can determine [OK]

Common Mistakes:

• Confusing closure with number of attributes
• Thinking closure is the primary key itself
• Ignoring functional dependencies in closure

2. Which of the following is the correct notation to represent the closure of attribute set {A, B}?

easy

A. {A, B}+

B. {A, B}*

C. closure(A, B)

D. closure{A, B}

Solution

1. Step 1: Recall standard closure notation

   The closure of a set of attributes X is denoted as X⁺, meaning all attributes functionally determined by X.

2. Step 2: Match notation to options

   {A, B}+ uses the correct notation {A, B}+. Other options are not standard or incorrect.

3. Final Answer:

   {A, B}+ -> Option A

4. Quick Check:

   Closure notation uses plus sign [OK]

Hint: Closure notation always uses plus sign after attribute set [OK]

Common Mistakes:

• Using star (*) instead of plus (+)
• Writing closure as a function call
• Misplacing braces or parentheses

3. Given relation R(A, B, C, D) and functional dependencies:
A -> B, B -> C, C -> D. What is the closure of {A}?

medium

A. {A, B}

B. {A, B, C}

C. {A, B, C, D}

D. {A}

Solution

1. Step 1: Start with attribute set {A}

   Initially, closure({A}) = {A}.

2. Step 2: Apply functional dependencies stepwise

   From A -> B, add B: closure = {A, B}. Then B -> C adds C: closure = {A, B, C}. Finally, C -> D adds D: closure = {A, B, C, D}.

3. Final Answer:

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

4. Quick Check:

   Closure includes all reachable attributes [OK]

Hint: Apply dependencies repeatedly until no new attributes added [OK]

Common Mistakes:

• Stopping after first dependency
• Ignoring transitive dependencies
• Confusing closure with original set

4. Identify the error in this closure calculation for relation R(A, B, C) with FDs:
A -> B, B -> C. Closure of {B} is given as {B, A}. What is wrong?

medium

A. Closure should include all attributes

B. Closure cannot include B itself

C. Closure should be empty

D. Closure should include C, not A

Solution

1. Step 1: Analyze given closure {B, A}

   Starting with {B}, applying B -> C adds C, so closure should be {B, C}.

2. Step 2: Check if A can be added

   There is no FD that allows B to determine A, so A should not be in closure.

3. Final Answer:

   Closure should include C, not A -> Option D

4. Quick Check:

   Closure only adds attributes functionally determined [OK]

Hint: Only add attributes on right side of FDs from closure set [OK]

Common Mistakes:

• Adding attributes not functionally dependent
• Confusing direction of dependencies
• Assuming closure includes all attributes

5. For relation R(A, B, C, D, E) with FDs:
A -> BC, CD -> E, B -> D. Which of the following is a candidate key?

hard

A. {A, D}

B. {A}

C. {A, C}

D. {B, C}

Solution

1. Step 1: Compute closure of {A}

   Start with {A}. From A -> BC, add B and C: closure = {A, B, C}. From B -> D, add D: closure = {A, B, C, D}. From CD -> E, since C and D are in closure, add E: closure = {A, B, C, D, E}.

2. Step 2: Check minimality of {A}

   Since closure({A}) includes all attributes, {A} is a superkey. It is minimal because it has only one attribute.

3. Final Answer:

   {A} -> Option B

4. Quick Check:

   Minimal set with full closure is candidate key [OK]

Hint: Check closure of smallest sets covering all attributes [OK]

Common Mistakes:

• Not checking transitive dependencies
• Assuming larger sets are always candidate keys
• Ignoring minimality condition