In relational database theory, Armstrong’s Axioms are a set of three basic rules that describe the logical behavior of functional dependencies. These rules are sound and complete, meaning all valid functional dependencies can be derived from them.

Armstrong’s Axioms are used to:

• Derive new functional dependencies from a given set.

• Compute attribute closure.

• Prove whether a functional dependency logically follows from a given set.

Three Basic Axioms

1. Reflexivity Rule

If $Y \subseteq X$, then $X \to Y$.

• Meaning: Any set of attributes functionally determines its own subsets.

• Such dependencies are trivial.

Example:
If $X = \{A, B\}$ and $Y = \{A\}$, then $\{A, B\} \to \{A\}$ holds.

2. Augmentation Rule

If $X \to Y$, then $XZ \to YZ$ for any attribute set $Z$.

• Meaning: Adding the same attributes to both sides of a functional dependency does not change its validity.

Example:
If $A \to B$, then $\{A, C\} \to \{B, C\}$.

3. Transitivity Rule

If $X \to Y$ and $Y \to Z$, then $X \to Z$.

• Meaning: Dependencies can be chained together.

Example:
If $A \to B$ and $B \to C$, then $A \to C$ follows.

Derived Rules

Using the three axioms, several useful derived rules can be obtained:

• Union Rule:
  If $X \to Y$ and $X \to Z$, then $X \to YZ$.

• Decomposition Rule:
  If $X \to YZ$, then $X \to Y$ and $X \to Z$.

• Pseudotransitivity Rule:
  If $X \to Y$ and $YZ \to W$, then $XZ \to W$.

These rules form the basis for algorithms that compute attribute closure and verify valid functional dependencies.

How Armstrong’s Axioms Help in Design

• They allow systematic derivation of all implied functional dependencies from a given set.

• They support normalization by helping to analyze dependencies.

• They provide a formal foundation for closure computation and key discovery.

Summary

Armstrong’s Axioms in DBMS are three basic rules—reflexivity, augmentation, and transitivity—that govern functional dependencies. From these, all valid functional dependencies can be derived. They are essential for understanding attribute closure, dependency reasoning, and normalization in relational database design.