Closure of attributes in DBMS Theory - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the closure of a set of attributes in a database?

The closure of a set of attributes is the complete set of attributes that can be functionally determined from the original set using the given functional dependencies.

beginner

Why is finding the closure of attributes important in database design?

It helps to find all attributes that depend on a given set, which is useful for checking keys and normalizing tables.

beginner

What is the first step in computing the closure of a set of attributes?

Start with the original set of attributes and add attributes that can be functionally determined using the functional dependencies until no more can be added.

intermediate

If the closure of an attribute set includes all attributes of a relation, what does that imply?

It means the attribute set is a candidate key for that relation.

intermediate

Can the closure of attributes change if functional dependencies change?

Yes, the closure depends on the functional dependencies. Changing them can add or remove attributes from the closure.

What does the closure of a set of attributes represent?

A Attributes unrelated to the set

B Only the original attributes in the set

C All attributes in the database

D All attributes functionally dependent on the set

The closure includes all attributes that can be functionally determined from the original set.

If the closure of attribute set X includes all attributes of relation R, what is X?

A A candidate key

B A foreign key

C A non-key attribute

D An irrelevant attribute

If X's closure covers all attributes, X is a candidate key.

Which of these is needed to compute the closure of attributes?

A Functional dependencies

B Primary key only

C Foreign keys only

D Data values

Functional dependencies define how attributes determine others, essential for closure.

What happens if you add more functional dependencies to the set when computing closure?

A Closure becomes empty

B Closure always shrinks

C Closure can only stay the same or grow

D Closure is unrelated to dependencies

More dependencies can add more attributes to the closure or keep it the same.

Why is closure useful in normalization?

A To store data faster

B To find keys and remove redundancy

C To encrypt data

D To create indexes

Closure helps identify keys and dependencies, which are important for normalization.

Explain the process of finding the closure of a set of attributes with an example.

Think about how you can find all attributes related to your starting set.

Describe why the closure of attributes is important for identifying candidate keys in a relation.

Consider what it means if your attribute set can determine everything.

Practice

1. What does the closure of attributes represent in database design?

easy

A. The total number of tables in a database

B. All attributes functionally determined by a given set of attributes

C. The list of all primary keys in a database

D. The count of foreign keys referencing a table

Solution

1. Step 1: Understand the concept of closure

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

2. Step 2: Relate closure to database design

   This helps identify all attributes dependent on a set, useful for normalization and design.

3. Final Answer:

   All attributes functionally determined by a given set of attributes -> Option B

4. Quick Check:

   Closure = all functionally determined attributes [OK]

Hint: Closure means all attributes you can find from given ones [OK]

Common Mistakes:

• Confusing closure with number of tables
• Thinking closure lists keys only
• Mixing closure with foreign key counts

2. Which of the following is the correct notation for the closure of attribute set A given functional dependencies F?

easy

A. A+

B. F+

C. A*

D. F*

Solution

1. Step 1: Recall standard notation

   The closure of an attribute set A under functional dependencies F is denoted as A+.

2. Step 2: Differentiate from other notations

   F+ denotes closure of the set of functional dependencies, not attributes. A* and F* are not standard notations here.

3. Final Answer:

   A+ -> Option A

4. Quick Check:

   Attribute closure = A+ notation [OK]

Hint: Closure of attributes is always written as A+ [OK]

Common Mistakes:

• Using F+ for attribute closure
• Confusing closure notation with star (*)
• Mixing attribute and dependency closures

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

medium

A. {A, B, C, D}

B. {A, B, C}

C. {A, B}

D. {A, D}

Solution

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

   Closure begins with {A}.

2. Step 2: Apply functional dependencies repeatedly

   From A -> B, add B; from B -> C, add C; from C -> D, add D.

3. Final Answer:

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

4. Quick Check:

   Applying all dependencies gives full set [OK]

Hint: Keep adding attributes until no new ones appear [OK]

Common Mistakes:

• Stopping after first dependency
• Forgetting to add transitive dependencies
• Confusing closure with just initial attributes

4. Identify the error in this closure calculation: Given A -> B, B -> C, and starting with {A}, the closure is calculated as {A, C}.

medium

A. Closure should be empty

B. Incorrect initial attribute set

C. Wrong functional dependencies used

D. Missed adding attribute B before C

Solution

1. Step 1: Check closure calculation steps

   Starting with {A}, applying A -> B adds B first.

2. Step 2: Verify transitive dependencies

   Only after B is added can B -> C add C. Skipping B is incorrect.

3. Final Answer:

   Missed adding attribute B before C -> Option D

4. Quick Check:

   Closure must add intermediate attributes [OK]

Hint: Add attributes step-by-step, don't skip intermediates [OK]

Common Mistakes:

• Skipping intermediate attributes
• Assuming direct jump to last attribute
• Ignoring order of dependencies

5. Given attributes A, B, C, D, E and functional dependencies:
A -> BC, CD -> E, B -> D. What is the closure of {A, D}?

hard

A. {A, C, D, E}

B. {A, B, C, D}

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

D. {A, B, C, E}

Solution

1. Step 1: Start with {A, D}

   Initial closure is {A, D}.

2. Step 2: Apply A -> BC

   Add B and C, closure becomes {A, B, C, D}.

3. Step 3: Apply B -> D

   D already included, no change.

4. Step 4: Apply CD -> E

   Since C and D are in closure, add E.

5. Final Answer:

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

6. Quick Check:

   All attributes reachable from {A, D} included [OK]

Hint: Use all dependencies repeatedly until no new attributes [OK]

Common Mistakes:

• Not applying CD -> E after adding C
• Forgetting to add B and C from A
• Stopping closure too early