DBMS Notes
HubUnit 1 — Introduction to DBMS & Entity-Relationship ModelAdvantages and Disadvantages of DBMSAggregation — Database Management SystemsCharacteristics of DBMSData IndependenceData ModelsDatabase Users and AdministratorsDBMS LanguagesER DiagramsEntity-Relationship (ER) ModelIntroduction to DBMSKeys in DBMSRelationships and CardinalitySchema and InstanceSpecialization and GeneralizationDBMS Unit 1 Summary - Database FundamentalsThree-Level Architecture (ANSI/SPARC Architecture)Weak Entity SetUnit 2 — Relational Model & SQLConstraints in SQLCursorsDCL and TCL CommandsDDL CommandsDML CommandsDomain Relational Calculus (DRC) - Complete TutorialFunctions and Stored ProceduresJoins in SQLNested Queries (Subqueries)Relational AlgebraRelational Calculus in Database Management SystemsRelational ModelSQL IntroductionDBMS Unit 2 Summary - SQL and Relational ModelTriggers in SQL — Database Management SystemsTuple Relational Calculus (TRC)Views in SQL — Database Management SystemsArmstrong\Boyce-Codd Normal Form (BCNF) — Database Management SystemsClosure of AttributesDependency PreservationFifth Normal Form (5NF)First Normal Form (1NF) - Database NormalizationFourth Normal Form (4NF)Functional DependencyLossless DecompositionMultivalued DependencyNormal Forms — OverviewNormalization — IntroductionSecond Normal Form (2NF) — Database Management SystemsUnit 3 Summary — NormalizationThird Normal Form (3NF) — Database Management SystemsUnit 4 — Transaction Management & Concurrency ControlACID PropertiesCheckpointsConcurrency Control & SerializabilityDeadlock in DBMSLock-Based Protocols & Two-Phase LockingLog-Based RecoveryRecovery SystemSerializabilityShadow PagingUnit 4 Summary — Transaction Management & Concurrency ControlTimestamp-Based ProtocolTransaction ManagementTransaction StatesTwo-Phase Locking (2PL)Unit 5 — Storage & IndexingB+ TreeB-TreeCost EstimationDynamic HashingFile OrganizationHashing TechniquesIndexing — IntroductionMulti-Level IndexingQuery Optimization — Database Management SystemsQuery ProcessingSingle-Level IndexingStatic HashingUnit 5 Summary — Storage & IndexingUnit 6 — Advanced Topics in DBMSBig Data and DBMSCloud DatabaseData MiningData WarehouseDatabase SecurityDistributed DatabaseDistributed Query ProcessingDistributed DBMS Architecture, Fragmentation & ReplicationMongoDB — IntroductionNoSQL DatabaseReplication in Distributed DatabasesUnit 6 Summary — Advanced Topics in DBMSViva Voce — DBMSImportant Viva Questions — DBMSInterview Questions — DBMSViva Answers — DBMSDBMS Multiple Choice Questions (MCQs)DBMS Lab Programs
Armstrong\
Armstrong's Axioms
Armstrong'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.
These axioms are sound (they only derive valid FDs) and complete (they can derive ALL FDs in F+).
Three Derived Rules (from the primary axioms)
4. Union
If X → Y and X → Z, then X → YZ.
Example
A → B and A → C ⟹ A → BC
5. Decomposition
If X → YZ, then X → Y and X → Z.
Example
A → BC ⟹ A → B and A → C
6. Pseudo-transitivity
If X → Y and WY → Z, then WX → Z.
Example
A → B and CB → D ⟹ CA → D
Closure of a Set of Attributes (X⁺)
The closure of attribute set X under functional dependency set F, written X⁺, is the set of all attributes that are functionally determined by X.
Algorithm to Compute X⁺
Algorithm: Closure(X, F)
1. result = X (start with X itself)
2. Repeat until no change:
For each FD (A → B) in F:
If A ⊆ result:
result = result ∪ B
3. Return result = X⁺
Worked Examples
Example 1
| F = { A | B, B → CD, A → E } |
| Step 0 | result = {A} |
| Step 1: A | B applies (A ⊆ {A}) → result = {A, B} |
| Step 2: B | CD applies (B ⊆ {A,B}) → result = {A, B, C, D} |
| Step 3: A | E applies (A ⊆ {A,B,C,D}) → result = {A, B, C, D, E} |
| A⁺ = {A, B, C, D, E} | A is a superkey (determines all attributes) |
Example 2
| F = { AB | C, C → D, D → A } |
| Step 0 | result = {A, B} |
| Step 1: AB | C applies → result = {A, B, C} |
| Step 2: C | D applies → result = {A, B, C, D} |
| Step 3: D | A applies → (A already in result) |
| (AB)⁺ = {A, B, C, D} | AB is a superkey |
| Step 0 | result = {C} |
| Step 1: C | D applies → result = {C, D} |
| Step 2: D | A applies → result = {A, C, D} |
| C⁺ = {A, C, D} | C is NOT a superkey (B not in C⁺) |
Example 3 — Finding Candidate Keys
| F = { AB | C, D → E, B → D, AB → E } |
| AB | C → {A,B,C} |
| B | D → {A,B,C,D} |
| D | E → {A,B,C,D,E} |
| (AB)⁺ = {A,B,C,D,E} | AB is a superkey |
| B | D → {B,D} |
| D | E → {B,D,E} |
Closure of a Set of FDs (F⁺)
The closure F⁺ is the set of ALL FDs that can be derived from F using Armstrong's Axioms. It includes both trivial and non-trivial dependencies.
| F = {A | B, B → C} |
| A | B (given) |
| B | C (given) |
| A | C (transitivity) |
| A | A (reflexivity) |
| A | BC (union + above) |
| AB | B (reflexivity) |
| AB | C (from A→C + augmentation) |
Checking FD Membership
To check if a specific FD X → Y is in F⁺:
- Compute X⁺ using F
- If Y ⊆ X⁺ then X → Y ∈ F⁺
| F = {A | B, B → C} |
| Is A | C in F⁺? |
| Compute A⁺: {A} | {A,B} → {A,B,C} |
| Is C ∈ A⁺? YES | A → C ∈ F⁺ ✓ |
Exam Focus
Revise definitions, diagrams, examples, and short-answer points for Armstrong\.
Interview Use
Prepare one clear explanation, one practical example, and one common mistake for this Database Management Systems (DBMS) topic.
Search Terms
dbms, database management systems (dbms), unit, armstrong, axioms, armstrong\
Related Database Management Systems (DBMS) Topics
Continue learning this concept
Unit 3Boyce-Codd Normal Form (BCNF) — Database Management SystemsA relation R is in BCNF (Boyce-Codd Normal Form) if, for every non-trivial functional dependency X → Y in R, the determinant X is a superkey of R. This is a...Unit 3Closure of AttributesThe closure of a set of attributes X (denoted X⁺) under a set of functional dependencies F is the set of ALL attributes that can be functionally determined...Unit 3Dependency PreservationA decomposition of relation R into R1, R2, ..., Rn is dependency-preserving if every functional dependency in the original set F can be verified using only...Unit 3Fifth Normal Form (5NF)A relation R is in Fifth Normal Form (5NF) — also called Project-Join Normal Form (PJ/NF) — if every non-trivial join dependency in R is implied by the...Unit 3First Normal Form (1NF) - Database NormalizationComplete guide to First Normal Form (1NF) normalization. Learn what 1NF is, why itUnit 3Fourth Normal Form (4NF)A relation R is in Fourth Normal Form (4NF) if, for every non-trivial multivalued dependency X →→ Y in R, the determinant X is a superkey of R. Fourth...