Introduction to database management systems

Relational database systems are expected to consist of a query language that can assist its users to query database instances. The two kinds of query languages: relational algebra and relational calculus.

Relational Algebra

Relational algebra is a procedural query language, which consists of a set of operations that take one or two relations as input and produce a new relation as result. The some of the basic operations are:

• Selection (σ): Selects a subset of rows from relation.
• Projection (π): Deletes unwanted columns from relation.
• Cross-Product (×): Allows us to combine two relations.
• Set-Difference (−): Tuples in reln. 1, but not in reln. 2.
• Union (U): Tuples in reln. 1 and in reln. 2

Projection 

Deletes attributes that are not in projection list.

Projection operator has to eliminate duplicates.

Notation − ∏A1, A2, An (r) 

Where A1, A2, An are attribute names of relation r.

Selection 

Selects rows that satisfy selection condition. 

Notation − σp(r) 

Cross-Product 

Each row of R1 is paired with each row of S1.

Set-Difference

The result of set difference operation is tuples, which is present in one relation not in second relation.

Notation − r – s

Union

The operations take two input relations, which should be union-compatible on: 

• Same number of fields.
• Fields have same type.

Relational Calculus

Relational calculus is nonprocedural type. There are two approaches: tuple relational calculus and domain relational calculus

Tuple Relational Calculus (TRC): Variables range over (i.e., get bound to) tuples. 

Notation − {T | Condition}

Domain Relational Calculus (DRC): Variables range over domain elements (= field values).

Notation − n{ a1, a2, a3, ..., an | P (a1, a2, a3, ... ,an)}

Where a1, a2 are attributes and P stands for formulae built by inner attributes

Both TRC and DRC are simple subsets of first-order logic