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 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⁺ ✓ |
Continue the lesson path
Study next in this module
01Unit 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...02Unit 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...03Unit 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...04Unit 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...