Relational Theory

The Relational Model

E.F. Codd introduced the relational model in his 1970 paper “A Relational Model of Data for Large Shared Data Banks.” The model provides a mathematically rigorous foundation for data management Based on set theory and first-order predicate logic. Every SQL database is an approximation of this Model — and understanding where SQL deviates from the theory helps you write correct queries.

Codd”s 12 Rules (and Rule 0)

Codd defined 13 rules (numbered 0 through 12) that a system must satisfy to be considered truly Relational. No commercial database fully satisfies all 13, but they serve as the theoretical Benchmark:

Rule	Name	Summary
0	Foundation	A relational DBMS must manage databases through its relational capabilities alone
1	Information	All information is represented as values in tables
2	Guaranteed Access	Every value is accessible by table name, primary key, and column name
3	Systematic Treatment of NULL	NULL values are distinct from default values and represent missing information
4	Dynamic Online Catalog	The database description (catalog) is represented as relational tables
5	Comprehensive Sublanguage	Supports at least one relational language (SQL, QBE, etc.)
6	View Updating	All views theoretically updatable must be updatable by the system
7	High-level Insert/Update/Delete	Set-level operations, not row-by-row processing
8	Physical Data Independence	Application logic unaffected by physical storage changes
9	Logical Data Independence	Application logic unaffected by logical schema changes (view changes)
10	Integrity Independence	Integrity constraints are part of the schema, not the application
11	Distribution Independence	Applications unaffected by data distribution
12	Nonsubversion	Low-level language cannot bypass integrity constraints

In practice, Rule 6 (view updating) is the most commonly violated. Most SQL databases cannot update Through arbitrary views, especially those involving joins, aggregations, or DISTINCT.

Relations, Tuples, Attributes, and Domains

The relational model uses precise terminology that SQL conflates:

Relational Term	SQL Term	Mathematical Object
Relation	Table	A set of tuples (no duplicate tuples)
Tuple	Row	An ordered list of values
Attribute	Column	A named position in a tuple
Domain	Data type	A set of permissible values
Degree	Arity	Number of attributes in a relation
Cardinality	Row count	Number of tuples in a relation

Key Distinction: Relations Are Sets

A mathematical relation is a set of tuples, which means:

No duplicate tuples — SQL tables allow duplicates (unless you declare UNIQUE or PRIMARY KEY). To get true relational behavior, you must use SELECT DISTINCT.
No ordering of tuples — SQL ORDER BY operates on the result set, not on the base relation. A table has no inherent row order.
Attributes are identified by name, not position — SQL allows SELECT * which relies on column ordering. This is a deviation from the theory.

Domains

A domain defines the set of valid values for an attribute. SQL data types (INTEGER VARCHAR(255)``DATE) are a coarse approximation of domains. A true domain would include Constraints:

CREATE DOMAIN age_domain AS INTEGER
  CHECK (VALUE >= 0 AND VALUE <= 150);

CREATE DOMAIN email_domain AS VARCHAR(255)
  CHECK (VALUE ~ '^[A-Za-z0-9._%+-]+@[A-Za-z0-9.-]+\.[A-Za-z]{2,}$');

PostgreSQL supports CREATE DOMAIN with CHECK constraints. Most other databases require you to Attach CHECK constraints directly to columns. Using domains centralises validation logic and Prevents inconsistency across tables.

Keys

Keys are the mechanism by which tuples are identified and relationships between relations are Established. Understanding the key hierarchy is essential for correct schema design.

Superkey

A superkey is any set of attributes that uniquely identifies a tuple. A superkey may contain Redundant attributes. For a relation Employee(emp_id, ssn, name, dept):

{emp_id} — superkey
{ssn} — superkey
{emp_id, name} — superkey (redundant: emp_id alone suffices)
{emp_id, ssn} — superkey

Candidate Key

A candidate key is a minimal superkey — removing any attribute destroys uniqueness. Every Relation has at least one candidate key (if it is a proper set).

{emp_id} — candidate key
{ssn} — candidate key
{emp_id, name} — NOT a candidate key (not minimal)

Primary Key

The primary key is the candidate key chosen by the database designer as the principal identifier. It Is a design choice, not a mathematical property. Convention:

Choose a stable, never-changing attribute (not email, not phone number)
Prefer single-column keys for simplicity
Use surrogate keys (auto-increment, UUID) when no natural candidate key exists

Foreign Key

A foreign key is a set of attributes in one relation that references the primary key of another Relation. It enforces referential integrity: every value of the foreign key must exist as a value of The referenced primary key (or be NULL).

CREATE TABLE orders (
    order_id   INTEGER PRIMARY KEY,
    customer_id INTEGER NOT NULL,
    total      NUMERIC(10,2) NOT NULL,
    FOREIGN KEY (customer_id) REFERENCES customers(customer_id)
        ON DELETE CASCADE
        ON UPDATE RESTRICT
);

Referential actions:

Action	Behavior
`CASCADE`	Propagate the delete/update to child rows
`RESTRICT`	Block the delete/update if child rows exist
`SET NULL`	Set the foreign key to NULL in child rows
`SET DEFAULT`	Set the foreign key to its default value in child rows
`NO ACTION`	Defer check to end of transaction (similar to RESTRICT in practice)

Composite Key

A composite key is a primary key consisting of two or more attributes. Common in association tables And many-to-many relationships:

CREATE TABLE enrollment (
    student_id  INTEGER NOT NULL,
    course_id   INTEGER NOT NULL,
    enrolled_at TIMESTAMP DEFAULT NOW(),
    PRIMARY KEY (student_id, course_id),
    FOREIGN KEY (student_id) REFERENCES students(student_id),
    FOREIGN KEY (course_id) REFERENCES courses(course_id)
);

Relational Algebra

Relational algebra provides a formal set of operations for manipulating relations. Every SQL query Is an expression in relational algebra (with extensions). Understanding these operations helps you Reason about query correctness and performance.

Selection (sigma)

Filters tuples based on a predicate. Denoted $\sigma_{\mathrm{predicate}(R)$ .

-- sigma_{salary > 100000}(Employee)
SELECT * FROM Employee WHERE salary > 100000;

Projection (pi)

Selects a subset of attributes. Denoted $\pi_{\mathrm{attr_1, \mathrm{attr_2}(R)$ .

-- pi_{name, salary}(Employee)
SELECT name, salary FROM Employee;

Cartesian Product (cross join)

Combines every tuple from one relation with every tuple from another. Denoted $R \times S$ .

SELECT * FROM Employee CROSS JOIN Department;

The Cartesian product of relations with $m$ and $n$ tuples produces $m \times n$ tuples. For tables With millions of rows, an accidental Cartesian product (missing JOIN condition) will produce Trillions of rows and exhaust memory.

Join

A join combines related tuples from two relations based on a join condition. The most common is the Natural join, which matches on attributes with the same name.

Join Type	Description
Inner Join	Returns tuples from both relations that satisfy the join condition
Left Outer	All tuples from left, matched tuples from right (NULL if no match)
Right Outer	All tuples from right, matched tuples from left (NULL if no match)
Full Outer	All tuples from both relations (NULL where no match)
Theta Join	General join with arbitrary condition $\theta$
Equi Join	Theta join where $\theta$ is equality on specific attributes

The inner join can be expressed in terms of selection and Cartesian product:

$R \bowtie_{\theta} S = \sigma_{\theta}(R \times S)$

Division

Division answers “which X values are associated with ALL Y values?” Denoted $R \div S$ .

Given Takes(student, course) and RequiredCourse(course)Find students who have taken ALL Required courses:

$\pi_{\mathrm{student, \mathrm{course}(\mathrm{Takes) \div \pi_{\mathrm{course}(\mathrm{RequiredCourse)$

SQL has no direct division operator. You simulate it with a double NOT EXISTS:

SELECT DISTINCT student
FROM Takes T1
WHERE NOT EXISTS (
    SELECT course FROM RequiredCourse R
    WHERE NOT EXISTS (
        SELECT 1 FROM Takes T2
        WHERE T2.student = T1.student
          AND T2.course = R.course
    )
);

Set Operations

Relational algebra includes standard set operations (relations must be union-compatible: same Degree and corresponding domains):

Union ( $R \cup S$ ): all tuples from both relations, duplicates removed
Intersection ( $R \cap S$ ): tuples present in both relations
Difference ( $R - S$ ): tuples in $R$ but not in $S$

SELECT name FROM customers
UNION
SELECT name FROM suppliers;

SELECT name FROM customers
INTERSECT
SELECT name FROM suppliers;

SELECT name FROM customers
EXCEPT
SELECT name FROM suppliers;

Rename (rho)

The rename operation, denoted $\rho_{\mathrm{new\_name}(R)$ Gives a relation or its attributes new Names. In SQL, this is AS:

SELECT E.name AS employee_name, M.name AS manager_name
FROM Employee E
JOIN Employee M ON E.manager_id = M.emp_id;

Functional Dependencies

A functional dependency (FD) is a constraint of the form $X \rightarrow Y$ Meaning “for any two Tuples, if they agree on all attributes in $X$ They must also agree on all attributes in $Y$ .” $X$ Is called the determinant.

Formally:

$X \rightarrow Y \iff \forall t_1, t_2 \in R : t_1[X] = t_2[X] \Rightarrow t_1[Y] = t_2[Y]$

Examples

For a relation Student(student_id, name, email, major, advisor_id, advisor_name):

student_id → name, email, major, advisor_id — student ID determines everything
advisor_id → advisor_name — advisor ID determines advisor name
student_id, course_id → grade — in an enrollment relation

Trivial vs Non-trivial

Trivial: $Y \subseteq X$ (e.g., {student_id} → {student_id}) — always true, provides no information
Non-trivial: $Y \not\subseteq X$ — carries actual semantic meaning
Completely non-trivial: $X \cap Y = \emptyset$

Armstrong’s Axioms

Armstrong’s axioms are a sound and complete set of inference rules for deriving all valid functional Dependencies from a given set:

Reflexivity: If $Y \subseteq X$ Then $X \rightarrow Y$
Augmentation: If $X \rightarrow Y$ Then $XZ \rightarrow YZ$ for any $Z$
Transitivity: If $X \rightarrow Y$ and $Y \rightarrow Z$ Then $X \rightarrow Z$

Derived rules (provable from the three axioms above):

Union: If $X \rightarrow Y$ and $X \rightarrow Z$ Then $X \rightarrow YZ$
Decomposition: If $X \rightarrow YZ$ Then $X \rightarrow Y$ and $X \rightarrow Z$
Pseudotransitivity: If $X \rightarrow Y$ and $YW \rightarrow Z$ Then $XW \rightarrow Z$

Attribute Closure

To determine if $X \rightarrow Y$ holds given a set of FDs $F$ Compute the closure of $X$ under $F$ Denoted $X^+$ :

Start with $X^+ = X$
Repeatedly apply Armstrong’s axioms to add attributes to $X^+$
Stop when no more attributes can be added
$X \rightarrow Y$ holds if and only if $Y \subseteq X^+$

Given R(A, B, C, D) with FDs: {A → B, B → C, C → D}

Closure of {A}:
  A⁺ = {A}
  A → B, so add B: A⁺ = {A, B}
  B → C, so add C: A⁺ = {A, B, C}
  C → D, so add D: A⁺ = {A, B, C, D}

Since A⁺ = {A, B, C, D}, A is a superkey.

Minimal Cover (Canonical Cover)

A minimal cover $F_{\min}$ of a set of FDs $F$ satisfies:

Every FD in $F_{\min}$ has a single attribute on the right side (decomposition)
Removing any FD from $F_{\min}$ changes the closure
Removing any attribute from the left side of any FD in $F_{\min}$ changes the closure

Algorithm:

Split right sides: replace $X \rightarrow YZ$ with $X \rightarrow Y$ and $X \rightarrow Z$
Remove redundant FDs: for each FD $f \in F$ Check if $(F - \{f\})^+ = F^+$ . If yes, remove $f$ .
Remove redundant attributes from left sides: for each FD $X \rightarrow Y$ and each attribute $A \in X$ Check if $(X - \{A\})^+ \supseteq Y$ . If yes, remove $A$ from $X$ .

Normalisation

Normalisation is the systematic process of organising a relation to eliminate redundancy and prevent Anomalies. Each normal form addresses specific types of redundancy.

Anomalies

Unnormalised data suffers from three types of anomalies:

Insertion anomaly: You cannot insert a fact without inserting unrelated facts (e.g., cannot add a new department without an employee)
Deletion anomaly: Deleting a fact unintentionally deletes other facts (e.g., deleting the last employee in a department removes the department)
Update anomaly: Updating a fact requires updating multiple rows (e.g., changing a department name requires updating every employee in that department)

First Normal Form (1NF)

A relation is in 1NF if and only if every attribute contains atomic (indivisible) values. No Repeating groups, no arrays, no nested relations.

Violates 1NF:
| student_id | name       | courses                    |
|------------|------------|----------------------------|
| 1          | Alice      | {Math, Physics, Chemistry} |

Satisfies 1NF:
| student_id | course     |
|------------|------------|
| 1          | Math       |
| 1          | Physics    |
| 1          | Chemistry  |

PostgreSQL arrays (INTEGER[]) technically violate 1NF but are a pragmatic extension. When you need To query individual elements or enforce referential integrity on array elements, model them as Separate rows.

Second Normal Form (2NF)

A relation is in 2NF if it is in 1NF and no non-prime attribute is partially dependent on any Candidate key. “Partially dependent” means dependent on a proper subset of a candidate key.

2NF only matters for relations with composite candidate keys. If all candidate keys are single Attributes, the relation is automatically in 2NF if it is in 1NF.

R(order_id, product_id, quantity, product_name)

FDs: {order_id, product_id} → quantity, product_id → product_name

product_name depends on product_id (a subset of the key), not the full key.
This violates 2NF.

Fix: split into:
  OrderItem(order_id, product_id, quantity)
  Product(product_id, product_name)

Third Normal Form (3NF)

A relation is in 3NF if it is in 2NF and no non-prime attribute is transitively dependent on any Candidate key. Equivalently, for every non-trivial FD $X \rightarrow A$ where $A$ is non-prime, $X$ Must be a superkey.

R(emp_id, name, dept_id, dept_name, dept_location)

FDs: emp_id → name, dept_id, dept_name, dept_location
     dept_id → dept_name, dept_location

dept_name and dept_location transitively depend on emp_id via dept_id.
This violates 3NF.

Fix: split into:
  Employee(emp_id, name, dept_id)
  Department(dept_id, dept_name, dept_location)

Boyce-Codd Normal Form (BCNF)

A relation is in BCNF if for every non-trivial FD $X \rightarrow Y$ , $X$ is a superkey. BCNF is Stricter than 3NF: 3NF allows $X \rightarrow A$ where $X$ is a superkey OR $A$ is a prime attribute. BCNF removes the “or $A$ is prime” exception.

R(student, course, instructor)

FDs: {student, course} → instructor
     instructor → course

Neither student nor instructor alone is a superkey.
The candidate keys are {student, course} and {student, instructor}.
instructor → course violates BCNF (instructor is not a superkey).

Fix: split into:
  Teaching(instructor, course)
  StudentInstructor(student, instructor)

Achieving BCNF may sometimes cause lossy decompositions (you cannot reconstruct the original Relation from the decomposed relations without losing information). In such cases, staying in 3NF is The practical compromise.

Fourth Normal Form (4NF)

A relation is in 4NF if it is in BCNF and contains no non-trivial multivalued dependencies. A Multivalued dependency $X \twoheadrightarrow Y$ means: for each value of $X$ The set of $Y$ values Is independent of the set of $Z$ values (where $Z = R - X - Y$ ).

R(emp_id, skill, language)

An employee has multiple skills and speaks multiple languages.
Skills and languages are independent: {emp_id} ⇸ {skill} and {emp_id} ⇸ {language}.

Violates 4NF if both MVDs hold and there are multiple skills AND languages.

Fix: split into:
  EmployeeSkill(emp_id, skill)
  EmployeeLanguage(emp_id, language)

Fifth Normal Form (5NF) / Project-Join Normal Form (PJNF)

A relation is in 5NF if it cannot be decomposed into smaller relations without losing information (no join dependency that is not implied by candidate keys). This is the “final” normal form in terms Of dependency theory.

In practice, 5NF violations are rare. They arise when a relation encodes a constraint that cannot be Expressed as a functional or multivalued dependency, such as a cyclic ternary relationship.

Denormalisation

Normalisation eliminates redundancy but can hurt query performance by requiring joins. In read-heavy Workloads, strategic denormalisation trades write complexity for read performance.

When to Denormalise

Read-heavy workloads where the same join is executed thousands of times per second
Reporting/OLAP where query flexibility matters more than write efficiency
Materialised views where the denormalised data is a derived, refreshable copy

Patterns

Precomputed aggregates: store total_order_value alongside order rows instead of computing it from order_items on every read
Embedded documents: in NoSQL stores, embed frequently accessed related data (e.g., order items inside the order document)
Copy fields: duplicate a frequently-filtered field (e.g., customer_name in the orders table) to avoid joins
Materialised views: precomputed query results that are refreshed on a schedule or trigger

Trade-offs

Normalised:
  + No update anomalies
  + Single source of truth
  + Schema is self-documenting
  - More joins (higher read latency)
  - More complex queries

Denormalised:
  + Fewer joins (lower read latency)
  + Simpler queries
  - Update anomalies (must update multiple locations)
  - Data inconsistency risk
  - More storage consumed

Denormalise from a position of knowledge. Start normalised, measure query performance, and Denormalise specific bottlenecks. Premature denormalisation creates maintenance burden that is far More expensive than the joins it eliminates.

ER Diagrams

Entity-Relationship (ER) diagrams are the standard notation for conceptual database design. They Model entities (things), attributes (properties), and relationships (associations between entities).

Notation

erDiagram
    CUSTOMER {
        int customer_id PK
        string name
        string email
        date created_at
    }

    ORDER {
        int order_id PK
        int customer_id FK
        numeric total
        timestamp created_at
    }

    ORDER_ITEM {
        int order_id FK
        int product_id FK
        int quantity
        numeric unit_price
    }

    PRODUCT {
        int product_id PK
        string name
        string category
        numeric price
    }

    CUSTOMER ||--o{ ORDER : "places"
    ORDER ||--|{ ORDER_ITEM : "contains"
    PRODUCT ||--o{ ORDER_ITEM : "referenced_in"

Cardinality Constraints

Notation	Meaning
1:1	Each entity relates to exactly one other
1:N	One entity relates to many others
M:N	Many entities relate to many others
0..1:1..N	Optional/mandatory cardinalities

Converting ER Diagrams to Relations

Strong entities become tables with their attributes as columns
Weak entities become tables that include the primary key of the identifying (owner) entity
1:1 relationships: add the foreign key to either table (prefer the table where NULL is less common)
1:N relationships: add the foreign key to the “N” side
M:N relationships: create an association table with composite primary key

Common Pitfalls

Using Natural Keys When Surrogate Keys Are Appropriate

A natural key like email address or national ID number may seem appealing, but it creates problems When the value changes (requiring cascading updates across all referencing tables) or when the Domain is not truly unique (two people share an email during a migration). Surrogate keys (BIGSERIAL``UUID) are stable, never change, and simplify joins.

Over-Normalising

Splitting tables too aggressively (e.g., creating a separate table for every attribute “just in Case”) makes even simple queries require joins. If a group of attributes always appears together and Is always updated together, they likely belong in the same table.

Ignoring Functional Dependencies in Application Code

If you find yourself writing application logic like “when the department changes, look up the new Department name and update this record,” you have a functional dependency that belongs in the Schema. Model it as a foreign key relationship.

Confusing NULL with Empty String or Zero

NULL means “unknown” or “not applicable.” It is not the same as '' (empty string) or 0. In SQL, NULL = NULL is NULL (not TRUE), and NULL + 1 is NULL. This three-valued logic causes Subtle bugs in WHERE clauses, JOIN conditions, and CHECK constraints.

Not Declaring Foreign Keys

Some teams omit foreign key constraints for “performance reasons.” The cost of a foreign key check On insert/update is negligible compared to the cost of orphaned rows, inconsistent data, and the Debugging time required to find them. Only omit foreign keys if you have a proven, measured reason And a compensating data integrity mechanism.

Using SELECT * in Application Code

SELECT * breaks when columns are added, reordered, or renamed. It transfers unnecessary data over The network. It prevents the query planner from using covering indexes. Always list the columns you Need explicitly.

Treating Multi-Valued Attributes as Single Columns

Storing comma-separated values, JSON arrays, or space-delimited lists in a single column violates 1NF and makes queries unreliable:

-- Anti-pattern:
CREATE TABLE articles (
    article_id INTEGER PRIMARY KEY,
    title VARCHAR(200),
    tags VARCHAR(500)  -- "database,performance,sql"
);

-- Querying requires string manipulation (fragile, slow, cannot use indexes):
SELECT * FROM articles WHERE tags LIKE '%database%';
-- Also matches "non-database", "database-admin", etc.

-- Correct: junction table
CREATE TABLE article_tags (
    article_id INTEGER REFERENCES articles(article_id),
    tag        VARCHAR(50) NOT NULL,
    PRIMARY KEY (article_id, tag)
);

CREATE INDEX idx_article_tags_tag ON article_tags(tag);

SELECT a.* FROM articles a
JOIN article_tags at ON a.article_id = at.article_id
WHERE at.tag = 'database';

Ignoring the Closure Algorithm

When designing schemas, developers often guess which functional dependencies exist rather than Computing them systematically. This leads to incorrect normal forms and redundant relationships. Always compute the attribute closure for candidate keys and verify the minimal cover before Declaring a schema to be in a given normal form.

Circular Foreign Key References

Two tables with foreign keys referencing each other create a circular dependency that complicates Inserts, deletes, and schema migrations:

-- Circular dependency:
CREATE TABLE employees (
    emp_id       INTEGER PRIMARY KEY,
    dept_id      INTEGER REFERENCES departments(dept_id)
);
CREATE TABLE departments (
    dept_id      INTEGER PRIMARY KEY,
    manager_id   INTEGER REFERENCES employees(emp_id)
);

-- Inserting requires DEFERRABLE constraints or inserting NULLs first
-- Dropping either table requires CASCADE

Break circular dependencies by making one side DEFERRABLE, using a NULL placeholder for the initial Insert, or redesigning the schema to eliminate the cycle.

Theoretical Foundations

Relational Calculus

Relational calculus is an alternative to relational algebra for expressing queries. It is declarative (describes what to retrieve, not how) and comes in two forms:

Tuple relational calculus: variables range over tuples.

$\{ t \mid \exists s \in \mathrm{Employee(s[\mathrm{dept] = t[\mathrm{dept] \land s[\mathrm{salary] \gt 100000) \}$

Domain relational calculus: variables range over attribute values (domains).

$\{ \lt \mathrm{name, \mathrm{salary \gt \mid \exists d, s (\mathrm{Employee(d, \mathrm{name, s) \land s \gt 100000) \}$

Codd’s theorem (1972) proves that relational algebra and relational calculus are equivalent in Expressive power: every query expressible in one is expressible in the other. SQL is based on Relational algebra with some relational calculus influences.

The Universal Relation Assumption

The universal relation assumption states that all attributes have globally unique names and that any Attribute can be related to any other. This assumption underlies many visual query tools and some ORM systems, but it does not hold in practice. Attributes named id``nameOr type appear in Many tables with completely different semantics.

Implication: always qualify column names with their table (or alias) in queries, and use descriptive Names that reflect the domain (e.g., customer_id instead of id).

Lossless-Join Decomposition

A decomposition of relation $R$ into $R_1$ and $R_2$ is lossless if $R_1 \bowtie R_2 = R$ (no Information is lost). A decomposition is lossless if and only if:

$R_1 \cap R_2 \rightarrow R_1 \mathrm{ or R_1 \cap R_2 \rightarrow R_2$

That is, the common attributes form a superkey for at least one of the decomposed relations.

R(A, B, C) with FDs: {A → B, B → C}

Decompose into R1(A, B) and R2(B, C):
  R1 ∩ R2 = {B}
  B → C (from the FDs), so B is a superkey for R2? No, R2 needs {B, C} and B → C, so B is a
  superkey for R2. Lossless decomposition. ✓

Decompose into R1(A, B) and R1(A, C):
  R1 ∩ R2 = {A}
  A → B (from the FDs), so A is a superkey for R1. Lossless decomposition. ✓

Dependency Preservation

A decomposition is dependency-preserving if every functional dependency in the original set can Be checked by examining only the decomposed relations (without joining them back together).

Not every lossless decomposition is dependency-preserving, and not every dependency-preserving Decomposition is lossless. The goal is to achieve both.

R(A, B, C) with FDs: {A → B, B → C}

Decompose into R1(A, B) and R1(A, C):
  Lossless? Yes (A → B, so A is a superkey for R1).
  Dependency-preserving?
    A → B: checkable on R1 ✓
    B → C: NOT checkable on either R1 or R2 alone ✗
  This decomposition loses the dependency B → C.

When a BCNF decomposition is not dependency-preserving, you have a choice: stay in 3NF (which always Has a dependency-preserving, lossless decomposition) or accept the non-dependency-preserving BCNF Decomposition and enforce the lost dependency through application logic or triggers.

Join Dependencies and 5NF Revisited

A join dependency (JD) is a generalisation of a multivalued dependency. A relation $R$ satisfies a Join dependency $JD(R_1, R_2, \ldots, R_n)$ if $R$ is equal to the join of its projections on $R_1, R_2, \ldots, R_n$ :

$R = R_1 \bowtie R_2 \bowtie \ldots \bowtie R_n$

5NF states that every non-trivial join dependency must be implied by candidate keys. When this Condition is violated, the relation encodes a constraint that is not captured by functional or Multivalued dependencies.

Classic 5NF example: supplier-part-project

R(supplier, part, project)

Business rule: if a supplier supplies a part and works on a project, then that supplier
supplies that part for that project. This is a ternary relationship that cannot be decomposed
into binary relationships without losing the constraint.

Supplier S supplies Part P and works on Project J:
  (S, P, J) must exist if (S, P) and (S, J) both exist.

Decomposing into SP(S, P), SJ(S, J), PJ(P, J) loses the ternary constraint.
The join SP ⋈ SJ ⋈ PJ may contain rows not in the original relation.

5NF violation: the constraint requires all three components together.

Practical Normalisation Workflow

The following workflow applies normalisation systematically to a real-world schema:

1. Identify all attributes and their semantics (what does each column mean?)
2. Identify all candidate keys (compute attribute closures)
3. List all functional dependencies (from business rules and key analysis)
4. Compute the minimal cover (canonical form)
5. Decompose into 1NF: ensure atomic values
6. Decompose into 2NF: eliminate partial dependencies on composite keys
7. Decompose into 3NF: eliminate transitive dependencies
8. Verify BCNF: check that every determinant is a superkey
9. For each decomposition, verify lossless-join and dependency-preservation
10. Evaluate denormalisation for known performance bottlenecks

Worked Example

Schema: R(student_id, course_id, instructor_name, instructor_office, grade)

Step 1: Identify attributes
  student_id       -- identifies a student
  course_id        -- identifies a course
  instructor_name  -- name of the instructor teaching the course
  instructor_office -- office of the instructor
  grade            -- grade the student received in the course

Step 2: Identify candidate keys
  {student_id, course_id} -- a student takes a course once

Step 3: Functional dependencies
  {student_id, course_id} → grade                    -- a student's grade for a course
  course_id → instructor_name, instructor_office     -- a course has one instructor

Step 4: 1NF? Yes, all attributes are atomic.

Step 5: 2NF? No partial dependencies:
  instructor_name and instructor_office depend on course_id (a subset of the key).
  They do NOT depend on student_id at all.

Step 6: Decompose into 2NF:
  R1(student_id, course_id, grade)  -- key: {student_id, course_id}
  R2(course_id, instructor_name, instructor_office)  -- key: {course_id}

Step 7: 3NF? Check transitive dependencies:
  R1: {student_id, course_id} → grade. No transitive dependency. 3NF. ✓
  R2: course_id → instructor_name → instructor_office? Only if one instructor
      has one office. If so, this is a transitive dependency.

  If instructor_name → instructor_office:
    R2a(course_id, instructor_name)
    R2b(instructor_name, instructor_office)

  If instructor_office is not functionally dependent on instructor_name
  (e.g., shared offices), R2 is already in 3NF.

Step 8: BCNF? All determinants are superkeys. ✓

This systematic approach prevents the common mistake of over-normalising (splitting tables that have No redundancy) or under-normalising (leaving transitive dependencies that cause update anomalies).

Summary

This topic covers the essential chemistry of relational theory, including key reactions, underlying theories, and practical applications.

Key concepts include:

key chemical principles and theories
mathematical relationships in chemistry
practical techniques and apparatus
applications of chemistry in industry
environmental and ethical considerations

Mastery of these concepts requires both theoretical understanding and the ability to apply knowledge to unfamiliar contexts, particularly in calculation and practical questions.

Worked Examples

Worked examples demonstrating the application of key concepts are covered in the detailed sub-pages linked above.

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