A Redundant Functional Dependency is a dependency that can be removed from the set of functional dependencies without affecting the overall closure of the set.
In simple words, a redundant FD is unnecessary because it can be derived from other functional dependencies.
A → B
B → C
A → C
F = { A → B, B → C, A → C }
Check if A → C is redundant:
Remove A → C and compute closure of A:
A⁺ = {A, B, C}
Since we still get C using A → B and B → C, the dependency A → C is redundant.
Final Set: A → B, B → C
Is redundant FD same as extraneous attribute?
Given:
F = { X → YZ, ZW → P, P → Z, W → XPQ, XYQ → YW, WQ → YZ }
Find redundant functional dependencies.
Remove it and compute X⁺:
X⁺ = {X}
Cannot derive Y or Z → NOT redundant ✅
Remove it and compute (ZW)⁺:
ZW⁺ = {Z, W}
Apply W → XPQ:
ZW⁺ = {Z, W, X, P, Q}
We can derive P → ZW → P is REDUNDANT ❌
Remove it and compute P⁺:
P⁺ = {P}
Cannot derive Z → NOT redundant ✅
No other FD gives XPQ → NOT redundant ✅
Remove it and compute (XYQ)⁺:
XYQ⁺ = {X, Y, Q}
Apply X → YZ:
XYQ⁺ = {X, Y, Q, Z}
Cannot derive W → NOT redundant ✅
Remove it and compute (WQ)⁺:
WQ⁺ = {W, Q}
Apply dependencies:
WQ⁺ = {W, Q, X, P, Z, Y}
We can derive both Y and Z → WQ → YZ is REDUNDANT ❌
Redundant Functional Dependencies:
ZW → P
WQ → YZ
Final Reduced Set:
F = { X → YZ, P → Z, W → XPQ, XYQ → YW }
An Extraneous Attribute is an attribute that is not required in a functional dependency. If removing an attribute does not change the meaning of the dependency, then it is called extraneous.
In simple words, extraneous attributes are unnecessary parts of a functional dependency. Removing them helps simplify the database design and is an important step in Minimal Cover and Normalization.
AB → C
A → C
An attribute in the left-hand side is extraneous if it can be removed and the dependency still holds.
AB → C
A → C
Since A alone determines C, B is extraneous.
A → C
If multiple attributes exist on the right side, we check them individually.
A → BC
Split into:
A → B
A → C
Then check if any dependency is unnecessary.
AB → C
A → C
C → D
Check AB → C:
A⁺ = {A, C, D}
Since A alone determines C, B is extraneous.
A → C
C → D
Does removing an attribute always change dependency?
Given Functional Dependencies:
F = { X → Z, XY → WP, XY → ZWQ, XZ → R }
Identify extraneous attributes and simplify the functional dependencies.
Single attribute on LHS → nothing to remove ✅
Check if X is extraneous:
Remove X → check Y⁺
Y⁺ = {Y}
Y cannot determine W or P → X is NOT extraneous ❌
Check if Y is extraneous:
Remove Y → check X⁺
X⁺ = {X, Z, R}
X cannot determine W or P → Y is NOT extraneous ❌
Split RHS:
XY → Z
XY → W
XY → Q
Check XY → Z:
Since X → Z already exists
👉 Z is redundant on RHS ❌
Check XY → W:
From XY → WP, we already get W
👉 W is redundant ❌
Check XY → Q:
No other FD gives Q → keep it ✅
Check if Z is extraneous:
Remove Z → check X⁺
X⁺ = {X, Z, R}
Since X already gives R → Z is extraneous ❌
X → R
Remaining dependencies:
X → Z
XY → WP
XY → Q
X → R
Extraneous removed:
| Extraneous Attribute | Redundant FD |
|---|---|
| Removes part of FD | Removes whole FD |
| Used in canonical cover | Used in simplification |
| Check attribute removal | Check FD derivation |