Armstrong's axioms in DBMS Theory - Cheat Sheet & Quick Revision

Recall & Review

beginner

What are Armstrong's axioms in database management?

Armstrong's axioms are a set of rules used to infer all functional dependencies on a database relation. They help in reasoning about data dependencies and normalization.

beginner

Name the three basic Armstrong's axioms.

The three basic Armstrong's axioms are:
1. Reflexivity
2. Augmentation
3. Transitivity

beginner

Explain the Reflexivity axiom with an example.

Reflexivity states that if set Y is a subset of set X, then X functionally determines Y (X → Y).
Example: If X = {A, B} and Y = {A}, then {A, B} → {A} holds true.

intermediate

What does the Augmentation axiom state?

Augmentation means if X → Y holds, then adding the same set Z to both sides preserves the dependency: XZ → YZ.
Example: If {A} → {B}, then {A, C} → {B, C}.

intermediate

Describe the Transitivity axiom with a real-life analogy.

Transitivity says if X → Y and Y → Z, then X → Z.
Analogy: If knowing your friend's phone number (X) lets you reach them (Y), and reaching them (Y) lets you get their address (Z), then knowing the phone number (X) lets you get the address (Z).

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

A Augmentation

B Transitivity

C Reflexivity

D Decomposition

Reflexivity states that if Y is a subset of X, then X → Y.

If X → Y holds, which axiom allows you to conclude that XZ → YZ also holds?

A Augmentation

B Union

C Transitivity

D Reflexivity

Augmentation allows adding the same set Z to both sides of a functional dependency.

Given X → Y and Y → Z, what can you infer using Armstrong's axioms?

A X → Z

B Z → X

C Y → X

D Z → Y

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

Which of the following is NOT one of the basic Armstrong's axioms?

A Augmentation

B Commutativity

C Transitivity

D Reflexivity

Commutativity is not a basic Armstrong's axiom.

Armstrong's axioms are primarily used for what purpose in databases?

A To manage user permissions

B To create tables

C To write SQL queries

D To infer functional dependencies

Armstrong's axioms help infer all functional dependencies in a relation.

Explain Armstrong's axioms and their importance in database normalization.

Think about how these rules help organize data efficiently.

Describe each of the three basic Armstrong's axioms with simple examples.

Use small sets of attributes to illustrate each axiom.

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