Armstrong's axioms in DBMS Theory - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to represent the Reflexivity rule of Armstrong's axioms.

DBMS Theory

SELECT * FROM dependencies WHERE attribute_set [1] attribute_set;

Drag options to blanks, or click blank then click option'

A≠

B⊆

C=

D⊇

Attempts:

3 left

💡 Hint

Common Mistakes

Using equality (=) instead of subset (⊆).

Using superset (⊇) which reverses the relationship.

The Reflexivity rule states that if Y is a subset of X, then X determines Y. So the attribute set must be a subset (⊆).

2fill in blank

medium

Complete the code to represent the Augmentation rule of Armstrong's axioms.

DBMS Theory

IF X -> Y THEN X[1]Z -> Y[2]Z

Drag options to blanks, or click blank then click option'

A+

B||

C,

D&

Attempts:

3 left

💡 Hint

Common Mistakes

Using logical operators like || or & which are not for attribute sets.

Using + which is arithmetic, not set concatenation.

Augmentation means adding the same set Z to both sides of the dependency. The comma represents concatenation of attribute sets.

3fill in blank

hard

Fix the error in the code representing the Transitivity rule of Armstrong's axioms.

DBMS Theory

IF X -> Y AND Y -> Z THEN X -> [1]

Drag options to blanks, or click blank then click option'

AW

BY

CX

DZ

Attempts:

3 left

💡 Hint

Common Mistakes

Using Y or X instead of Z in the conclusion.

Using unrelated attribute W.

Transitivity states that if X determines Y and Y determines Z, then X determines Z.

4fill in blank

hard

Complete the code to complete the Union rule of Armstrong's axioms.

DBMS Theory

IF X -> Y AND X -> [1] THEN X -> Y,[1]

Drag options to blanks, or click blank then click option'

AZ

BY

C+

D,

Attempts:

3 left

💡 Hint

Common Mistakes

Using Y instead of Z in the second dependency.

Using + which is arithmetic, not set union.

The Union rule states that if X determines Y and X determines Z, then X determines the union of Y and Z. The comma represents combining attribute sets.

5fill in blank

hard

Fill both blanks to complete the Decomposition rule of Armstrong's axioms.

DBMS Theory

IF X -> [1],Y THEN X -> [2] AND X -> Y

Drag options to blanks, or click blank then click option'

AY

B,

C+

DZ

Attempts:

3 left

💡 Hint

Common Mistakes

Using + instead of comma to separate attributes.

Using Y instead of Z for the first attribute in the combined set.

The comma separates attribute sets.

Practice

1. Which of the following is NOT one of Armstrong's axioms used in database theory?

easy

A. Reflexivity

B. Augmentation

C. Transitivity

D. Normalization

Solution

1. Step 1: Recall Armstrong's axioms

   Armstrong's axioms include Reflexivity, Augmentation, and Transitivity.

2. Step 2: Identify the option not in the axioms

   Normalization is a database design process, not an axiom.

3. Final Answer:

   Normalization -> Option D

4. Quick Check:

   Armstrong's axioms exclude Normalization [OK]

Hint: Remember the three axioms: Reflexivity, Augmentation, Transitivity [OK]

Common Mistakes:

• Confusing normalization with axioms
• Thinking all database terms are axioms
• Mixing up augmentation with normalization

2. Which Armstrong's axiom states that if Y is a subset of X, then X functionally determines Y?

easy

A. Reflexivity

B. Transitivity

C. Augmentation

D. Decomposition

Solution

1. Step 1: Understand the axiom definitions

   Reflexivity states if Y is subset of X, then X determines Y.

2. Step 2: Match the statement to the axiom

   The given statement matches Reflexivity exactly.

3. Final Answer:

   Reflexivity -> Option A

4. Quick Check:

   Subset implies functional dependency by Reflexivity [OK]

Hint: Subset implies dependency = Reflexivity [OK]

Common Mistakes:

• Confusing reflexivity with transitivity
• Choosing decomposition which is not an axiom
• Mixing augmentation with reflexivity

3. Given the functional dependencies: A -> B and B -> C, what can we infer using Armstrong's axioms?

medium

A. A -> C

B. B -> A

C. C -> A

D. C -> B

Solution

1. Step 1: Identify the given dependencies

   We have A determines B, and B determines C.

2. Step 2: Apply Transitivity axiom

   Transitivity says if A -> B and B -> C, then A -> C.

3. Final Answer:

   A -> C -> Option A

4. Quick Check:

   Transitivity gives A -> C [OK]

Hint: Chain dependencies to find indirect ones using Transitivity [OK]

Common Mistakes:

• Reversing dependencies incorrectly
• Ignoring transitivity rule
• Assuming dependencies go backward

4. Identify the error in this application of Armstrong's axioms: Given A -> B, the conclusion is B -> A.

medium

A. Correct use of augmentation

B. Proper use of reflexivity

C. Mistakenly reversed dependency

D. Valid transitivity application

Solution

1. Step 1: Analyze the given dependency

   A -> B means A determines B, not the other way around.

2. Step 2: Check the conclusion

   Concluding B -> A reverses the dependency, which Armstrong's axioms do not allow.

3. Final Answer:

   Mistakenly reversed dependency -> Option C

4. Quick Check:

   Dependencies do not reverse automatically [OK]

Hint: Dependencies don't reverse unless explicitly stated [OK]

Common Mistakes:

• Assuming dependencies are symmetric
• Confusing augmentation with reversal
• Ignoring direction of functional dependency

5. If you have A -> B and A -> C, which Armstrong's axiom allows you to combine these into A -> BC?

hard

A. Transitivity

B. Union

C. Decomposition

D. Augmentation

Solution

1. Step 1: Understand the given dependencies

   A determines B and A determines C separately.

2. Step 2: Apply the Union axiom

   Union allows combining A -> B and A -> C into A -> BC.

3. Final Answer:

   Union -> Option B

4. Quick Check:

   Union combines dependencies into one [OK]

Hint: Combine dependencies with same left side using Union [OK]

Common Mistakes:

• Confusing union with augmentation
• Using decomposition instead of union
• Applying transitivity incorrectly here