Candidate key finding using closure in DBMS Theory - Step-by-Step Execution

Concept Flow - Candidate key finding using closure

Start with attribute set S

↓

Compute closure S+ using FDs

↓

Check if S+ contains all attributes

↓

Candidate key

↓

End

Start with a set of attributes, find its closure using functional dependencies, check if closure covers all attributes to identify candidate keys.

Execution Sample

DBMS Theory

Attributes: {A,B,C,D}
FDs: A->B, B->C, C->D
Check closure of {A}

Find closure of attribute set {A} to see if it covers all attributes, indicating a candidate key.

Analysis Table

Step	Current Closure	FD Applied	New Attributes Added	Closure After Step
1	{A}	A->B	B	{A,B}
2	{A,B}	B->C	C	{A,B,C}
3	{A,B,C}	C->D	D	{A,B,C,D}
4	{A,B,C,D}	No more FDs	None	{A,B,C,D}

💡 Closure contains all attributes {A,B,C,D}, so {A} is a candidate key.

State Tracker

Variable	Start	After 1	After 2	After 3	Final
Closure	{A}	{A,B}	{A,B,C}	{A,B,C,D}	{A,B,C,D}

Key Insights - 2 Insights

Why do we add new attributes to the closure only when the left side of an FD is in the closure?

Because only when the FD's left side attributes are in the closure can we infer the right side attributes, as shown in steps 1-3 of the execution_table.

What if the closure does not contain all attributes after applying all FDs?

Then the attribute set is not a candidate key; we must try a different set, as indicated by the 'No' branch in the concept_flow.

Visual Quiz - 3 Questions

Look at the execution_table, what is the closure after step 2?

A {A,B}

B {A,B,C}

C {A,B,C,D}

D {A}

💡 Hint

Check the 'Closure After Step' column for step 2 in the execution_table.

At which step does the closure first include attribute D?

A Step 1

B Step 2

C Step 3

D Step 4

💡 Hint

Look at the 'New Attributes Added' column to see when D is added.

If the FD C->D was missing, what would be the final closure of {A}?

A {A,B,C}

B {A,B}

C {A,B,C,D}

D {A}

💡 Hint

Consider the closure steps without applying C->D from the execution_table.

Concept Snapshot

Candidate key finding using closure:
- Start with attribute set S
- Compute closure S+ by applying FDs repeatedly
- If S+ contains all attributes, S is a candidate key
- Otherwise, try different attribute sets
- Closure helps find minimal keys covering all attributes

Full Transcript

To find candidate keys using closure, start with a set of attributes. Compute its closure by adding attributes that can be functionally determined using the given functional dependencies. Repeat until no new attributes can be added. If the closure contains all attributes of the relation, the starting set is a candidate key. If not, try another set. For example, with attributes A,B,C,D and FDs A->B, B->C, C->D, the closure of {A} is computed stepwise adding B, then C, then D, resulting in {A,B,C,D}. Since this closure contains all attributes, {A} is a 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