BE - Database Management Systems

Armstrong’s axioms are a set of three fundamental rules in database theory used to derive all possible functional dependencies from a given set. These rules help ensure the consistency and correctness of relationships within a database.

The Three Primary Armstrong’s Axioms

Reflexivity

If a set of attributes ($A$) contains another set ($B$), then $A$ determines $B$.

• Example: If you have roll_no and name, then you know name if you have both.

Augmentation

If $A$ determines $B$, then adding extra attributes ($Y$) to both sides will also hold: $AY\to BY$.

• Example: If roll_no → name, then roll_no, age → name, age also holds.

Transitivity

If $A\to B$ and $B\to C$, then $A\to C$.

• Example: If roll_no → name and name → age, then roll_no → age.

Armstrong’s Axioms Matter

• They allow logical reasoning about functional dependencies.

• They are sound (only valid dependencies can be generated) and complete (all valid dependencies can be found using them).

Example of All Three

Suppose you know:

• A → B

• B → C

By transitivity: A → C
By augmentation: A, D → B, D
By reflexivity: A → A

Armstrong’s axioms form the foundation for normalization and efficient relational database design.