Armstrong's axioms in DBMS Theory - Step-by-Step Execution

Concept Flow - Armstrong's axioms

Start with given FDs

↓

Apply Reflexivity

↓

Apply Augmentation

↓

Apply Transitivity

↓

Add new FDs

↓

Repeat until no new FDs

↓

Closure of FDs

Armstrong's axioms are a set of rules applied repeatedly to find all functional dependencies implied by a given set.

Execution Sample

DBMS Theory

Given FDs: A -> B, B -> C
Apply Transitivity: A -> C

Shows how to derive new functional dependencies using Armstrong's axioms.

Analysis Table

Step	Rule Applied	Input FDs	New FD Derived	Explanation
1	Given	A -> B, B -> C	None	Start with initial functional dependencies
2	Transitivity	A -> B, B -> C	A -> C	Since A determines B and B determines C, A determines C
3	Reflexivity	A -> B, B -> C, A -> C	A -> A	Every attribute determines itself
4	Augmentation	A -> B	AC -> BC	Adding attribute C to both sides of A -> B
5	Augmentation	B -> C	AB -> AC	Adding attribute A to both sides of B -> C
6	No new FDs	All above	None	No further dependencies can be derived
Exit	Termination			No new functional dependencies found, closure complete

💡 No new functional dependencies can be derived, so the process stops.

State Tracker

Functional Dependencies	Start	After Step 2	After Step 3	After Step 4	After Step 5	Final
FD Set	A->B, B->C	A->B, B->C, A->C	A->B, B->C, A->C, A->A	A->B, B->C, A->C, A->A, AC->BC	A->B, B->C, A->C, A->A, AC->BC, AB->AC	A->B, B->C, A->C, A->A, AC->BC, AB->AC

Key Insights - 3 Insights

Why do we add A->A using Reflexivity when it seems obvious?

Reflexivity states every attribute determines itself; including A->A ensures completeness and helps in further derivations as shown in step 3.

How does Augmentation work when adding attributes to both sides?

Augmentation adds the same attribute(s) to both sides of a FD, preserving the dependency; see steps 4 and 5 where C or A is added to both sides.

When do we stop applying Armstrong's axioms?

We stop when no new functional dependencies can be derived, as shown in step 6 and the exit note.

Visual Quiz - 3 Questions

Look at the execution table at step 2, what new functional dependency is derived?

A A -> C

B B -> A

C C -> A

D C -> B

💡 Hint

Check the 'New FD Derived' column in step 2 of the execution table.

At which step is the Augmentation rule first applied?

A Step 3

B Step 4

C Step 2

D Step 5

💡 Hint

Look for the first occurrence of 'Augmentation' in the 'Rule Applied' column.

According to the variable tracker, what is the FD set after step 5?

A A->B, B->C, A->C

B A->B, B->C, A->C, A->A

C A->B, B->C, A->C, A->A, AC->BC, AB->AC

D A->B, B->C

💡 Hint

Check the 'After Step 5' column in the variable tracker for the FD set.

Concept Snapshot

Armstrong's axioms are rules to find all functional dependencies:
1. Reflexivity: If B ⊆ A, then A -> B
2. Augmentation: If A -> B, then AC -> BC
3. Transitivity: If A -> B and B -> C, then A -> C
Apply repeatedly until no new dependencies appear.

Full Transcript

Armstrong's axioms are a set of three rules used in database management to find all functional dependencies implied by a given set. The rules are Reflexivity, Augmentation, and Transitivity. Starting with initial functional dependencies, we apply these rules step-by-step to derive new dependencies. For example, from A->B and B->C, we derive A->C by Transitivity. We also add trivial dependencies like A->A by Reflexivity. Augmentation allows adding attributes to both sides of a dependency. This process repeats until no new dependencies can be found, resulting in the closure of the functional dependencies.