Relational Algebra Overview

Relational Algebra

• A theoretical language used for modeling data in relational databases.
• Consists of operations that manipulate relations without altering the original data.
• Introduced by Edgar F. Codd while at IBM, forming the foundation for query languages like SQL.

Fundamental Operations

• Selection (Restriction): Filters tuples based on a specified condition.
• Projection: Extracts specific attributes from a relation while eliminating duplicates.
• Cartesian Product: Combines every tuple from two relations.
• Union: Merges two relations, including all unique tuples from both.
• Set Difference: Returns tuples from one relation that are not in another.
• Additional operations include Join, Intersection, and Division, expressible through fundamental operations.

Unary Operations

• Selection and Projection: Operate on a single relation to extract information.

Selection

• Notated as σpredicate(R), it yields tuples from relation R that meet a condition.
• SQL equivalent uses WHERE clause in select statements, handling three possible truth values (true, false, unknown).

Projection

• Notated as Πa1..., an(R), it generates a subset of R's columns while removing duplicates.

Union

• Notated as R ∪ S; combines tuples from relations R and S, ensuring they are union-compatible.

Set Difference

• Notated as R - S; defines tuples in R that are absent in S, requiring union compatibility.

Intersection

• Notated as R ∩ S; identifies tuples common to both R and S, with union compatibility required.

Cartesian Product

• Notated as R × S; produces a relation formed by pairing each tuple of R with every tuple of S.

Join Operations

• Types include:
  • Theta Join
  • Equijoin (a specific Theta join)
  • Natural join
  • Outer join
  • Semijoin

Theta Join

• Notated as R ⋈FS; produces tuples from the Cartesian product satisfying a specified predicate.

EquiJoin

• A specific case of Theta Join where the predicate involves only equality.

Natural Join

• Notated as R ⋈ S; merges relations R and S on all common attributes, resulting in a single occurrence of shared attributes.

Outer Join

• Notated as R ⋊ S; includes tuples from R that lack matches in S, filling gaps with null values.

Semijoin

• Notated as R ▹F S; returns tuples from R based on participation in the join satisfying a condition.

Division

• Notated as R ÷ S; forms a relation over attributes C that corresponds to all tuple combinations matching those in S.

Aggregate Functions

• COUNT: Counts the number of values in an attribute.
• SUM: Returns the total of numeric attributes (details not provided).