Understanding Soundness and Completeness of Armstrong’s Axioms

Question:

For a set of functional dependencies (FDs) F, F_AA represents the set of FDs derived using Armstrong’s Axioms, and F⁺ represents the closure of F. Considering the following statements:

• S₁: F_AA ⊆ F⁺
• S₂: F⁺ ⊆ F_AA

Which of the following is correct?

1. S₁ indicates the soundness of Armstrong’s Axioms, while S₂ indicates their completeness.
2. S₁ indicates the completeness of Armstrong’s Axioms, while S₂ indicates their soundness.
3. Both S₁ and S₂ demonstrate the soundness of Armstrong’s Axioms.
4. Both S₁ and S₂ demonstrate the completeness of Armstrong’s Axioms.

Solution:

Armstrong’s Axioms consist of a set of rules for deriving valid functional dependencies. The two fundamental properties are:

• Soundness: All FDs derived using Armstrong’s Axioms belong to the closure (F⁺), represented by S₁: F_AA ⊆ F⁺.
• Completeness: All FDs in the closure can be derived using Armstrong’s Axioms, represented by S₂: F⁺ ⊆ F_AA.

Correct Answer: Option 1: S₁ indicates the soundness of Armstrong’s Axioms, while S₂ indicates their completeness.

Explanation of Options:

• Option 1: Correct, as S₁ represents soundness and S₂ represents completeness.
• Option 2: Incorrect, as it reverses the meanings of S₁ and S₂.
• Option 3: Incorrect, as soundness is only represented by S₁, not S₂.
• Option 4: Incorrect, as completeness is only represented by S₂, not S₁.

Candidate Key Verification Using Closure

Question:

To confirm whether a set of attributes X constitutes a candidate key for a relation R, which of the following is used to ensure that R can be derived from X?

Options:

1. Attributes
2. Closure
3. Functional Dependencies
4. Prime Attributes

Correct Answer:

Closure

Solution:

The closure of a set of attributes X, represented as X⁺, includes all attributes that can be functionally derived from X using the functional dependencies of the schema. To determine if X is a candidate key:

1. Compute the closure X⁺ by starting with X and adding all directly derivable attributes.
2. Continue adding attributes that can now be derived until no new attributes can be added.
3. If X⁺ equals all attributes in R, then X is a candidate key.

This approach provides a systematic way to verify whether a set of attributes forms a key for the relation.

Explanation of Options:

• Option 1: Attributes are essential components but do not confirm key validity.
• Option 2: Correct, as closure ensures derivation of all attributes in the relation.
• Option 3: Functional dependencies define attribute relationships but do not confirm keys.
• Option 4: Prime attributes are part of candidate keys but are unrelated to closure calculation.

Functional Dependency Closure (\( F^+ \))

Question:

From the functional dependency set \( F = \{ A \to B, B \to C, C \to D \} \), what is the complete closure \( F^+ \), derived by applying Armstrong’s axioms?

Options:

1. \( \{ A \to B, B \to C, C \to D, A \to BCD \} \)
2. \( \{ A \to B, B \to C, C \to D, A \to C, B \to D, A \to D, B \to CD, A \to BCD \} \)
3. \( \{ A \to C, B \to D, A \to D \} \)
4. \( \{ A \to C, B \to D, A \to D, A \to BCD \} \)

Correct Answer:

Option 2

Solution:

To compute \( F^+ \), the closure of the functional dependency set \( F \), we repeatedly apply Armstrong’s axioms (reflexivity, augmentation, and transitivity) to infer all possible functional dependencies.

Step-by-Step Derivation:

1. Start with \( F = \{ A \to B, B \to C, C \to D \} \).
2. Using transitivity:
   • \( A \to B \) and \( B \to C \) imply \( A \to C \).
   • \( A \to C \) and \( C \to D \) imply \( A \to D \).
   • \( B \to C \) and \( C \to D \) imply \( B \to D \).
3. Using augmentation:
   • \( A \to B \) with \( B \to C \) and \( C \to D \) imply \( A \to BCD \).
   • \( B \to C \) and \( C \to D \) imply \( B \to CD \).

Final Closure (\( F^+ \)):

\( \{ A \to B, B \to C, C \to D, A \to C, B \to D, A \to D, B \to CD, A \to BCD \} \)

Explanation of Incorrect Options:

• Option 1: Omits key transitive and augmented dependencies such as \( A \to C, A \to D, B \to CD \).
• Option 3: Ignores dependencies \( A \to B \), \( A \to BCD \), and \( B \to CD \).
• Option 4: Partially correct but does not include \( B \to CD \) and other critical augmentations.

Functional Dependency Closure and Armstrong’s Axioms

Question

Given the functional dependency set:

F = { A → B, B → C, C → D }

Determine the closure F⁺, which is the complete set of functional dependencies that can be derived using Armstrong’s axioms.

Options

1. { A → B, B → C, C → D, A → BCD }
2. { A → B, B → C, C → D, A → C, B → D, A → D, B → CD, A → BCD }
3. { A → C, B → D, A → D }
4. { A → C, B → D, A → D, A → BCD }

Correct Answer

Option 2: { A → B, B → C, C → D, A → C, B → D, A → D, B → CD, A → BCD }

Solution

The closure F⁺ represents all functional dependencies that can be derived using Armstrong’s axioms.

Step-by-Step Explanation

1. Given: F = { A → B, B → C, C → D }
2. Apply Transitivity:
   • A → B and B → C imply A → C.
   • A → C and C → D imply A → D.
   • B → C and C → D imply B → D.
3. Apply Augmentation:
   • A → B, B → C, and C → D imply A → BCD.
   • B → C and C → D imply B → CD.

Final Closure (F⁺):

{ A → B, B → C, C → D, A → C, B → D, A → D, B → CD, A → BCD }

Explanation of Incorrect Options

• Option 1: Missing transitive dependencies like A → C, B → CD, A → D.
• Option 3: Incomplete; misses A → B, B → C, B → CD, A → BCD.
• Option 4: Incorrectly excludes B → CD.

Functional Dependency Inference using Armstrong’s Axioms

Question

If F = { AB → CD }, determine which of the following functional dependencies can be derived using Armstrong’s axioms.

Options

1. ABE → CDE
2. AB → CDE
3. AB → C
4. A → CD

Correct Answers

1. ABE → CDE

2. AB → C

Solution

We apply Armstrong’s axioms to derive valid functional dependencies:

Step-by-Step Explanation

1. Augmentation Rule:
   • From AB → CD, adding E to both sides gives ABE → CDE.
2. Decomposition Rule:
   • From AB → CD, we can infer AB → C and AB → D.
3. Invalid Dependencies:
   • AB → CDE: Cannot be directly inferred as AB → CD implies AB → C and AB → D separately.
   • A → CD: Cannot be inferred, as AB → CD does not imply A → CD without B.

Final Derived Functional Dependencies:

• ABE → CDE
• AB → C

Classifying Functional Dependencies (Trivial and Non-Trivial)

Question

Given A, B, C are attributes (note: they are not sets of attributes) of a relation, and consider the following functional dependencies:

• S₁: A → AB
• S₂: ABC → AB

Which of the following options correctly classifies S₁ and S₂?

Options

1. S₁ is a trivial functional dependency (FD) and S₂ is a non-trivial FD.
2. S₁ is a non-trivial FD and S₂ is a trivial FD.
3. Both S₁ and S₂ are trivial FDs.
4. Both S₁ and S₂ are non-trivial FDs.

Correct Answer

2. S₁ is a non-trivial FD and S₂ is a trivial FD.

Solution

Step-by-Step Explanation:

1. Definition of a Trivial Functional Dependency (FD):
   • A functional dependency X → Y is trivial if Y ⊆ X.
2. Analyzing S₁: A → AB:
   • Here, Y = AB, and X = A.
   • Since Y ⊄ X (AB ⊄ A), S₁ is a non-trivial FD.
3. Analyzing S₂: ABC → AB:
   • Here, Y = AB, and X = ABC.
   • Since Y ⊆ X (AB ⊆ ABC), S₂ is a trivial FD.

Conclusion:

• S₁ is a non-trivial FD.
• S₂ is a trivial FD.

Computational Aspects of Functional Dependencies

Question

Consider the following statements regarding computational aspects of functional dependencies (FDs):

1. S₁: Computing the closure of a set of functional dependencies F⁺ is computationally more expensive than computing the attribute closure X⁺_F.
2. S₂: Computing F⁺ is computationally less expensive than computing X⁺_F.
3. S₃: The FDs that hold on a relation can be determined from the data present in the relation.
4. S₄: The FDs that hold on a relation cannot be determined from the data present in the relation.

Which of the following options is correct?

Options

1. S₁ and S₃ are true.
2. S₁ and S₄ are true.
3. S₂ and S₃ are true.
4. S₂ and S₄ are true.

Correct Answer

Option 2: S₁ and S₄ are true.

Solution

Step-by-Step Explanation:

1. Analyzing S₁:
   • Computing the closure F⁺ involves determining all possible FDs derivable from a given set F, which is computationally intensive as it requires examining all subsets of attributes and their possible closures.
   • On the other hand, computing X⁺_F (the closure of a specific attribute set X) is localized to a single subset of attributes and is relatively less computationally expensive.
   • Hence, S₁ is true.
2. Analyzing S₂:
   • As per the above reasoning, S₂ contradicts S₁ and is therefore false.
3. Analyzing S₃:
   • The FDs that hold on a relation are determined by the schema design and not purely by the data present in the relation. For instance, data instances may accidentally satisfy an FD that does not logically hold.
   • Hence, S₃ is false.
4. Analyzing S₄:
   • The above reasoning supports S₄ as correct since FDs cannot be conclusively determined from the data.

Conclusion: S₁ and S₄ are correct statements.

Functional Dependency Derivation

Question

If F = { X → Y, WY → Z }, then which of the following is entailed by F?

Options

1. W → Z
2. WX → Z
3. X → Z
4. Y → Z

Correct Answer

Option 2: WX → Z

Solution

Step-by-Step Explanation:

1. Functional Dependencies Given:
   • F = { X → Y, WY → Z }
2. Objective:
   • To determine which of the given options can be derived using the functional dependencies provided.
3. Analysis of Each Option:
   • W → Z: Using the given dependencies, W → Z cannot be derived because neither W nor WY alone contains enough information to imply Z.
   • WX → Z:
     • WX implies WY because X → Y is given, and W is already present.
     • WY → Z is given in F.
     • Therefore, WX → Z can be derived.
     • Hence, this option is true.
   • X → Z: X → Z cannot be derived as X alone does not imply WY, which is required to imply Z.
   • Y → Z: Y alone cannot imply Z because WY → Z requires W in conjunction with Y.
4. Conclusion: Only WX → Z can be derived from the given dependencies.

Attribute Closure in Functional Dependencies

Question

For a relation R(A, B, C, D, E), the functional dependencies are { AB → C, B → D, C → E, D → A }. Which of the following represents B⁺ (attribute closure of B)?

Options

1. { B, D }
2. { B, D, A }
3. { B, D, A, C, E }
4. { B }

Correct Answer

Option 3: { B, D, A, C, E }

Solution

Step-by-Step Explanation:

1. What is Attribute Closure?
   • Attribute closure (X⁺) of a set of attributes X is the set of attributes that can be functionally determined by X using the given functional dependencies.
2. Functional Dependencies Given:
   • AB → C
   • B → D
   • C → E
   • D → A
3. Steps to Compute B⁺:
   • Start with B⁺ = { B }.
   • Apply B → D: Add D to B⁺, so B⁺ = { B, D }.
   • Apply D → A: Add A to B⁺, so B⁺ = { B, D, A }.
   • Apply AB → C: Since A and B are in B⁺, add C, so B⁺ = { B, D, A, C }.
   • Apply C → E: Add E to B⁺, so B⁺ = { B, D, A, C, E }.
4. Final Attribute Closure of B: B⁺ = { B, D, A, C, E }.
5. Verification of Options:
   • Option 1 ({ B, D }): Incomplete, as A, C, and E are missing.
   • Option 2 ({ B, D, A }): Incomplete, as C and E are missing.
   • Option 3 ({ B, D, A, C, E }): Correct, as it matches the computed closure.
   • Option 4 ({ B }): Incorrect, as it includes only B and ignores other attributes derived from B.

Deriving Functional Dependencies Using Armstrong’s Axioms

Question

Complete the following statement: In general, a functional dependency X → Y can be derived from F using Armstrong’s Axioms if and only if:

Options

1. Y ⊆ X⁺
2. Y ⊆ X
3. X ⊆ Y⁺
4. X ⊆ Y

Correct Answer

Option 1: Y ⊆ X⁺

Solution

Step-by-Step Explanation:

1. Functional Dependency Definition:
   • A functional dependency X → Y means that the attributes in Y are uniquely determined by the attributes in X.
2. Armstrong’s Axioms:
   • Reflexivity: If Y ⊆ X, then X → Y.
   • Augmentation: If X → Y, then XZ → YZ (where Z is a set of attributes).
   • Transitivity: If X → Y and Y → Z, then X → Z.
3. Attribute Closure X⁺:
   • X⁺ is the set of all attributes that can be functionally determined by X using the dependencies in F.
4. Condition for Deriving X → Y:
   • A functional dependency X → Y can be derived from F if all attributes in Y are included in the closure of X, i.e., Y ⊆ X⁺.
5. Verification of Options:
   • Option 1 (Y ⊆ X⁺): Correct. This is the precise condition for deriving X → Y.
   • Option 2 (Y ⊆ X): Incorrect. Y ⊆ X implies reflexivity but does not cover general derivations.
   • Option 3 (X ⊆ Y⁺): Incorrect. This does not determine if Y is functionally dependent on X.
   • Option 4 (X ⊆ Y): Incorrect. This implies the reverse relationship, which is unrelated to functional dependency derivation.

Functional Dependencies and Logical Implications

Question

Given a relation R(A, B, C, D, E) and the set of functional dependencies:

F = { A → B, A → C, CD → E, B → D, E → A }

Which of the following is not logically implied by F?

Options

1. CD → AC
2. BD → CD
3. EA → BD
4. DE → BC

Correct Answer

Option 2: BD → CD

Solution

Step-by-Step Explanation:

1. Understanding Logical Implication of Functional Dependencies:
   • A functional dependency X → Y is logically implied by a set F if Y ⊆ X⁺, where X⁺ is the closure of X under F.
2. Given Functional Dependencies:
   • F = { A → B, A → C, CD → E, B → D, E → A }
3. Checking Each Option:
   • Option 1 (CD → AC):
     • From CD → E and E → A, we can derive CD → A.
     • From CD → A and A → C, we can derive CD → AC.
     • Logically implied by F.
   • Option 2 (BD → CD):
     • From B → D, we can derive BD → D. However, there is no dependency that allows BD → CD.
     • Not logically implied by F.
   • Option 3 (EA → BD):
     • From E → A and A → B, we can derive EA → B.
     • From EA → B and B → D, we can derive EA → BD.
     • Logically implied by F.
   • Option 4 (DE → BC):
     • From D → B and E → A, we can derive DE → BA.
     • From A → C, we can derive DE → BC.
     • Logically implied by F.