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.