Relational Algebra

Introduction

Relational algebra is a procedural query language used to query and manipulate data in relational databases.

It provides a foundation for SQL, and defines operations on relations (or tables) to produce new relations.

The key features are:

• Operates on one or more relations.
• Outputs a new relation.
• Uses a mathematical approach.

It is very important because it:

• Forms the theoretical foundation of relational databases.
• Helps in query optimization.
• Translates user-friendly SQL queries into executable instructions.
• Promotes a deeper understanding of database operations.

Query Processing

Query processing steps include:

1. Parsing and translation
2. Optimization
3. Evaluation

Basic Relational Algebra Operations

Operation	Symbol	Example
SELECT	$\sigma$	$\sigma$ Status = “Active”(Customer)
PROJECTION	$\pi$	$\Pi$ CustomerName, Status (Customer)
RENAME	$\rho$	$\rho$ ${new_name} /${old_name}
UNION	$\cup$	$A \cup B$ gives all tuples in table $A$ and $B$.
INTERSECTION	$\cap$	$A \cap B$ gives all tuples both in $A$ and $B$.
DIFFERENCE	$-$	$A - B$ gives tuples in $A$ but not in $B$.
CARTESIAN PRODUCT	$\times$	$A \times B$ gives combinations of columns in $A$ and $B$.
JOIN	$\Join$	None

Modification of Database

Operations of modifying databases:

• Deletion
• Insertion
• Updating

These operations are expressed using the assignment operator $\gets$.

Deletion

$$r \gets r - E$$

Example:

$$\text{Customer} \gets \text{Customer} - (\sigma \; \text{Status = `Active'(Customer)})$$

Insertion

$$r \gets r \cup E$$

Example:

$$Customer \gets Customer \cup \{1, `\text{Name}', `\text{Active}'\}$$

Updating

Updating can be expressed by a sequence of deletion and insertion operations.