Relational Algebra - Expression and Operators

1 - About

Relational algebra defines the relational database through a set of data operators (select, filter, join, sort, union, etc.) that defines an intermediate format for query planning/optimization.

Information requests may be expressed using set notions and set operations.

Relational algebra is about Data Flow focusing on transformations (relational operator)

Because Relational expressions (operator) process data, their names are typically verbs:

  • Sort,
  • Join,
  • Project,
  • Filter,
  • Scan,
  • Sample.

Merge with Sql Engine - Relational Operator (Data operations|Execution Plan Steps) ?

Relational algebra is an extension to mathematical set theory (set operation), it was devised by E.F. Code (IBM) in 1970.


Programs that manipulate tabular data exhibit an algebraic structure, therefore:

  • It allows a reasoning and manipulation independently of physical data representation.
  • All operations on a table return a table.
  • The operation can then be chained together.

Relational algebra expressions dictate how to achieve an answer by giving what operations to do and in what order to do them. A declarative language only expresses conditions that must be met in order for a result to be an answer, not how to get that answer.

For instance, SQL is the what not the how. It's clear what we want unclear how to get it

Every query returns a table. The language is algebraically closed A view is stored as an algebraic closure so it can be optimized when used with queries.

3 - Order

By design, query results in relational algebra are unordered. Repeating the same query multiple times is not guaranteed to return results in the same order. Similarly, database-backed relational data also do not guarantee row order.

4 - Column

The column names of an operator are guaranteed to be unique.

In case of name conflict, a suffix is generally applied. For example, joining EMP to DEPT in a SCOTT schema, will result in

  • a column called DEPTNO
  • and another called DEPTNO_1.

5 - Relational Operator Algebra

All relational algebra operator are set-operator.

operator in Algebra.

Type Name Symbol Sql Operator Description
Normal select, Selection <math>\sigma</math> of s where select the rows
Normal project <math>\pi</math> select select the columns
Normal join <math>\Join</math> join join
Normal union <math>\cup</math> union (without duplicates)
of union all (with duplicates)
union of tables - Tuples either in Rel 1 or in Rel 2
Normal intersection <math>\cap</math> minus or join same row in two tables
Normal Set-difference - except, minus Tuples in Rel 1, but not Rel 2
Normal Rename <math>\rho</math> as Rename an attribute or a relation
Extended Grouping and aggregating g
Extended Duplicate elimination d
Extended Sorting t
Extended Division / find values that have all properties

6 - Bag vs Set

Relational Algebra has two semantics:

  • Set semantics = standard Relational Algebra
  • Bag semantics = extended Relational Algebra

Rule of thumb:

  • Every paper will assume set semantics
  • Every implementation will assume bag semantics

7 - Flow

Data Flow

Input Query Language Output
Sets of Tuples Set Relational Algebra Set of Tuples
Bags of Tuples Bag Relational Algebra Bag of Tuples
Lists of Tuples Extended Relational Algebra List of Tuples

8 - Algebraic Optimization

9 - Equivalent relational expressions with different costs

10 - Library

11 - Documentation / Reference

Data Science
Data Analysis
Data Science
Linear Algebra Mathematics

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