Closure of attributes in DBMS Theory - Practice Problems & Coding Challenges

Challenge - 5 Problems

🧠 Conceptual

intermediate

Understanding Attribute Closure

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

A {A, B, C, D}

B {A, B, C}

C {A, B, D}

D {A, B}

💡 Hint

Remember that closure includes all attributes you can reach using the functional dependencies starting from the given set.

The closure of S = {A, B} includes A and B themselves. Using A → C, we add C. Using B → D, we add D. So, S⁺ = {A, B, C, D}.

📋 Factual

intermediate

Properties of Attribute Closure

Which of the following statements about attribute closure is always true?

A The closure of an attribute set is always equal to the entire set of attributes in the relation.

B The closure of an attribute set never contains attributes outside the relation.

C The closure of an attribute set always contains the original set.

D The closure of an attribute set can be empty.

💡 Hint

Think about what closure means in terms of starting attributes.

By definition, the closure of an attribute set includes the attributes in the original set plus any attributes that can be functionally determined from it. So, it always contains the original set.

🔍 Analysis

advanced

Calculating Closure with Multiple Dependencies

Given relation R(A, B, C, D, E) and functional dependencies:

1. A → B
2. B → C
3. C → D
4. D → E

What is the closure of {A}?

A {A, B, C}

B {A, B, C, D, E}

C {A, B, C, D}

D {A, B}

💡 Hint

Apply the dependencies step-by-step starting from A.

Starting with {A}, apply A → B to add B. Then B → C adds C. Next, C → D adds D. Finally, D → E adds E. So closure is {A, B, C, D, E}.

❓ Comparison

advanced

Comparing Closures of Different Attribute Sets

Given relation R(A, B, C) with functional dependencies:
1. A → B
2. B → C

Which attribute set has the larger closure?

A Closures of {A} and {B} are equal

B Closure of {B} is larger than closure of {A}

C Closures of {A} and {B} contain only their own attributes

D Closure of {A} is larger than closure of {B}

💡 Hint

Calculate both closures step-by-step.

Closure({A}) = {A, B, C} because A → B and B → C. Closure({B}) = {B, C} because B → C. So closure of {A} is larger.

❓ Reasoning

expert

Determining Candidate Keys Using Closure

Relation R(A, B, C, D) has functional dependencies:
1. A → B
2. B → C
3. C → A

Which of the following attribute sets is a candidate key for R?

A {A, D}

B {D}

C {C, D}

D {B, D}

💡 Hint

Check which sets' closure includes all attributes of R.

Closure({A, D}) includes A, B (A→B), C (B→C), and D itself, so it covers all attributes. Similarly, {B, D} and {C, D} also cover all attributes due to cyclic dependencies among A, B, C. But {D} alone does not. However, minimality matters: {A, D} is minimal and a candidate key.