**Inference Rule (IR) in DBMS:**

- Armstrong's axioms in database management systems were
developed by
**William w. Armstrong**in 1974. - Armstrong's axioms are the basic
inference rule, used to conclude functional dependencies on a relational
database.
- It provides a set of rules for a
simple reasoning technique in functional dependencies.
- An inference rule is an
assertion that can apply a user on a set of functional dependencies to
derive other FD (functional dependencies).

Inference
rules are divided into major two parts:-

I.
Axioms or
primary rules.

II.
Additional
rules or secondary rules.

**I.
****Axioms or primary
rules.**

**1.
****Reflexive Rule (IR**_{1})

_{1})

**2.
****Augmentation Rule (IR**_{2})

_{2})

**3.
****Transitive Rule (IR _{3})**

** **

**II.
****Additional
rules or secondary rules.**

**1.
****Union Rule (IR _{4})**

**2.
****Decomposition Rule (IR**_{5})

_{5})

**3.
****Pseudo transitive Rule (IR**_{6})

_{6})

The
Functional dependency has 6 types of inference rules:

**1. Reflexive Rule (IR**_{1})

_{1})

If X is a set of attributes and Y is the subset of X, then X
functionally determines Y.

In the
reflexive rule, if Y is a subset of X, then X determines Y.

1.
If X ⊇ Y then X → Y

**Example:**

1.
X = {a, b, c, d, e}

- Y = {a, b, c}

**2. Augmentation Rule (IR**_{2})

_{2})

The
augmentation is also called a partial dependency. If X determines Y, then XZ
determines YZ for any Z.

if X determines Y and Z is any attribute set, then XZ determines YZ. It
is also called a partial dependency.

1.
If X → Y then
XZ → YZ

**Example:**

1.
For R(ABCD), **if** A → B then AC → BC

**3. Transitive Rule (IR**_{3})

_{3})

In the
transitive rule, if X determines Y and Y determines Z, then X must also
determine Z.

if X determines Y and Y determines Z, then X also determines Z.

1.
If X → Y and Y → Z then X → Z

**4. Union Rule (IR**_{4})

_{4})

This rule is also known as the **additive rule**. if X determines Y and X
determines Z, then X also determines both Y and Z.

Union rule
says, if X determines Y and X determines Z, then X must also determine Y and Z.

1.
If X → Y and X → Z then X → YZ

**Proof:**

1. X → Y (given)

2. X → Z (given)

3. X → XY (using IR_{2} on 1 by augmentation with X. Where XX = X)

4. XY → YZ (using IR_{2} on 2 by augmentation with Y)

5. X → YZ (using IR_{3} on 3 and 4)

**5. Decomposition Rule (IR**_{5})

_{5})

This rule is the reverse of the Union rule and is also known as the **project rule**.

if X determines Y and Z together, then X determines Y and Z separately

1.
If X → YZ then X → Y and X → Z

**Proof:**

1. X → YZ (given)

2. YZ → Y (using IR_{1} Rule)

3. X → Y (using IR_{3} on 1 and 2)

**6. Pseudo transitive Rule (IR**_{6})

_{6})

In the **pseudo
transitive rule**, if X determines Y, and YZ determines W, then XZ also determines W.

1.
If X → Y and YZ → W then XZ → W

**Proof:**

1. X → Y (given)

2. WY → Z (given)

3. WX → WY (using IR_{2} on 1 by augmenting with W)

4. WX → Z (using IR_{3} on 3 and 2)

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