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Closure of Attributes
Last Updated : 21 May, 2026
The closure of a set of attributes X denoted X⁺ under a set of functional dependencies F is the set of ALL attributes that are functionally determined by X.
Unit 3180 words12 headingsTables includedExamples included
Definition
The closure of a set of attributes X (denoted X⁺) under a set of functional dependencies F is the set of ALL attributes that are functionally determined by X.
X⁺ = { A | X → A can be derived from F using Armstrong's Axioms }
Algorithm
| Input | Attribute set X, FD set F |
| Output | X⁺ (closure of X) |
| FOR each FD (LHS | RHS) in F DO |
Applications of Attribute Closure
1. Check if X is a Superkey
X is a superkey if X⁺ contains all attributes of R.
R(A,B,C,D), F = {A→B, B→C, C→D}
A⁺ = {A,B,C,D} = all attributes → A is a superkey ✓
2. Find All Candidate Keys
A candidate key is a minimal superkey.
For each subset X of R, compute X⁺.
If X⁺ = R (all attributes), X is a superkey.
If no proper subset of X is also a superkey, X is a candidate key.
3. Check if an FD X → Y Holds
X → Y holds if and only if Y ⊆ X⁺.
4. Find Canonical Cover
Used to eliminate redundant FDs and redundant attributes from FD left-hand sides.
Detailed Worked Examples
Example 1 — Basic Closure
| F = { A | BC, CD → E, B → D, E → A } |
| Step 0 | result = {A, E} |
| A | BC → result = {A, B, C, E} |
| E | A → A already present |
| B | D → result = {A, B, C, D, E} |
| CD | E → E already present |
| (AE)⁺ = {A, B, C, D, E} | AE is a superkey |
| A⁺: {A} | A→BC → {A,B,C} → B→D → {A,B,C,D} → CD→E → {A,B,C,D,E} |
| A⁺ = {A,B,C,D,E} | A alone is a superkey! |
Example 2 — Finding All Candidate Keys
| F = { PQ | R, R → S, S → P } |
| P⁺: {P} — no FD starts with P alone | P⁺ = {P} ✗ not superkey |
| Q⁺: {Q} — no FD starts with Q alone | Q⁺ = {Q} ✗ |
| R⁺: {R} | R→S → {R,S} → S→P → {P,R,S} ✗ (Q missing) |
| S⁺: {S} | S→P → {P,S} ✗ |
| (PQ)⁺: {P,Q} | PQ→R → {P,Q,R} → R→S → {P,Q,R,S} ✓ SUPERKEY |
| Check subsets P⁺={P} ✗, Q⁺={Q} ✗ | PQ is a CANDIDATE KEY |
| (QR)⁺: {Q,R} | R→S → {Q,R,S} → S→P → {P,Q,R,S} ✓ SUPERKEY |
| Check Q⁺={Q} ✗, R⁺={P,R,S} ✗ | QR is a CANDIDATE KEY |
| (QS)⁺: {Q,S} | S→P → {P,Q,S} → no more → {P,Q,S} ✗ |
| Candidate Keys | {PQ}, {QR} |
Example 3 — Verifying an FD
| F = { A | B, B → C, C → D } |
| Q: Does A | D hold (i.e., is A → D in F⁺)? |
| {A} | A→B → {A,B} → B→C → {A,B,C} → C→D → {A,B,C,D} |
| D ∈ A⁺? YES | A → D holds ✓ |
Summary Table
| Use Case | How to Use Closure |
|---|---|
| Is X a superkey? | X⁺ = all attributes? |
| Is X → Y derivable? | Y ⊆ X⁺? |
| Find candidate keys | Find minimal X where X⁺ = all attributes |
| Check FD redundancy | Remove FD, recheck if closure is same |
| Check attribute redundancy | Remove attribute from LHS, recheck closure |
Exam Focus
Revise definitions, diagrams, examples, and short-answer points for Closure of Attributes.
Interview Use
Prepare one clear explanation, one practical example, and one common mistake for this DBMS topic.
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Unit 3Armstrong's Axioms & Closure of AttributesArmstrong's Axioms are a set of inference rules used to derive all functional dependencies FDs from a given set F. They were proposed by William W. Armstrong in 1974.Unit 3Boyce-Codd Normal Form BCNFA relation R is in BCNF Boyce-Codd Normal Form if, for every non-trivial functional dependency X → Y in R:Unit 3Dependency PreservationA decomposition of R into R1, R2, ..., Rn is dependency-preserving if the union of the functional dependencies that can be derived from each Ri restricted to the attributUnit 3Fifth Normal Form 5NFA relation R is in Fifth Normal Form 5NF — also called Project-Join Normal Form PJ/NF — if, for every non-trivial join dependency R1, R2, ..., Rn in R:Unit 3First Normal Form 1NFA relation is in First Normal Form 1NF if:Unit 3Fourth Normal Form 4NFA relation R is in Fourth Normal Form 4NF if, for every non-trivial multivalued dependency X →→ Y in R: