Relational Algebra in Database Management System

Formal Languages - I
(Relational Algebra)
10/09/2024
Christalin Nelson | Systems | SoCS
1 of 67

At a Glance
• Formal Languages
• Unary Relational Operations
• Operations from Set Theory
• Binary Relational Operations
• Additional Relational Operations
10/09/2024
2 of 66

Formal languages (1/2)
• The following formal languages are associated with Relational model
– Relational algebra
• Used in internals of many DB implementations for query processing and optimization
– Relational calculus
• Has firm basis on mathematical logic
• SQL for RDBMSs has some of its foundations in tuple relational calculus
10/09/2024
3 of 66

Formal languages (2/2)
• “Expressive power” of both languages are identical
– Any retrieval that is specified in basic relational algebra can also be specified in
relational calculus, and vice versa.
• “Relationally complete” language
– A relational query language (L) is relationally complete if any query that is expressed in
relational calculus can be expressed in L
– Important basis for comparing the expressive power of high-level query languages
– Note
• Most relational query languages are relationally complete but have more expressive power
than relational algebra or relational calculus because of additional operations (like
aggregate functions, grouping, and ordering)
10/09/2024
4 of 66

Relational Algebra (1/2)
• Procedural in nature
• Focuses on describing a sequence of operations to retrieve or manipulate data
– It is also possible to nest algebra operations to form a single expression
• Provides Set of operations that act on relations to produce new relations as results
– E.g. SELECT, PROJECT, JOIN, UNION, INTERSECTION, and DIFFERENCE
– Note:
• Relational algebra operations can be thought of as similar to SQL query clauses
– Relational algebra operations are expressed in a more formal and procedural manner
– Queries can be expressed as sequences of these operations, which are applied to relations to
obtain the desired result
10/09/2024
5 of 66

Relational Algebra (2/2)
• Types of Relational algebra operations
– Set operations from mathematical set theory
• Example: UNION, INTERSECTION, SET DIFFERENCE, CARTESIAN PRODUCT (CROSS
PRODUCT)
– Operations developed specifically for relational databases
• Example: SELECT, PROJECT, JOIN, DIVISION
– Additional Operations
• Aggregate functions (can summarize data from the tables)
• Additional types of JOIN and UNION operations (known as OUTER JOINs and OUTER
UNIONs)
10/09/2024
6 of 66

Unary Relational Operations: SELECT (1/5)
• Unary Operations
– Applied to a single relation (Example: SELECT, PROJECT)
• SELECT Operation
– Selects/Filters a subset of tuples from a relation that satisfies a selection condition
• i.e. Horizontal partition of relation
– Denoted by sigma
• <selection condition>
– Condition is applied independently to each tuple in relation (R). If TRUE => tuple is selected
– Contains Boolean expression which has clauses of the form
» <attribute name> <comparison operator> <constant value or attribute name>
10/09/2024
7 of 66

10/09/2024
8 of 67

• Boolean operators {AND, OR, NOT}
• Comparison operators {=, <, ≤, >, ≥, ≠}
– Note: If the domain of the attribute is
• Set of ordered values => All comparison operators can be used
• Set of unordered values => Only the comparison operators in the set {=, ≠} can be used
– Example: domain Color = { ‘red’, ‘blue’, ‘green’, ‘white’, ‘yellow’, ...}, where no order is specified
among the various colors
• Degree of resulting relation
– No. of attributes resulting from a SELECT operation
• For any condition, No. of tuples in resulting relation ≤ No. of tuples in R
– i.e. |σc (R)| ≤ |R|
10/09/2024
9 of 66

• Selectivity
– Fraction of tuples selected by a selection condition
• SELECT operation is commutative
– A sequence of SELECTs can be applied in any order
• Cascade/Combine
– SELECT operations can be cascaded into a single operation with an AND condition
10/09/2024
10 of 66

• In SQL, the SELECT condition is typically specified in the WHERE clause of a query
• Note:
– The SELECT operation is different from the SELECT clause of SQL
– The SELECT operation chooses tuples from a table and is sometimes called a RESTRICT
or FILTER operation
10/09/2024
11 of 66

Unary Relational Operations: PROJECT (1/3)
• Selects columns from the table and discards the other columns
– i.e. Vertical partition of relation (Recollect: SELECT did horizontal partition of relation)
• Denoted by pi
• Degree of resulting relation
– The result of PROJECT operation has only the attributes specified in <attribute list> in
the same order as they appear in the list
• Hence, Degree = Number of attributes in <attribute list>
• Duplicate elimination
– If the <attribute list> includes only non-key attributes of R, the result of PROJECT
operation is a set of distinct tuples => A valid relation
10/09/2024
12 of 66

10/09/2024
13 of 67

Unary Relational Operations: PROJECT (3/3)
• Number of Tuples in the resulting relation
– If (projection list includes some key of R)
• No. of tuples in the resulting relation = No. of tuples in R
– Else
• No. of tuples in a resulting relation ≤ No. of tuples in R
• The following expression is valid as long as <list2> contains the attributes in <list1>
• Commutativity does not hold on PROJECT
• In SQL, the PROJECT condition is typically specified in the SELECT clause of a query
10/09/2024
14 of 66

Sequences of Operations and RENAME Operation (1/5)
• Dealing with the Sequence of Operations
– Operations can be nested into a single relational algebra expression (Inline expression)
(or)
– One operation can be applied at a time with intermediate resulting relations
• i.e. Give names to the relations that hold the intermediate results
• Example: Retrieve first name, last name, salary of all employees who work in
department no. 5
– This needs PROJECT and SELECT operations to be sequenced
10/09/2024
15 of 66

10/09/2024
16 of 67

Using intermediate relations and renaming of attributes
10/09/2024
17 of 67

• Rename attributes in intermediate results
• RENAME operation
– Denoted as rho
– Types
• Rename both relation & attributes=>
• Rename the relation only =>
• Rename the attributes only =>
10/09/2024
18 of 66

• In SQL, a single query typically represents a complex relational algebra expression.
Renaming in SQL is accomplished by aliasing using AS
– Example: Renaming both relation and attributes
10/09/2024
19 of 66

Relational Algebra Operations from Set Theory (1/13)
• Binary operations are used to work with the elements of two sets in various ways
• Classifications
– Set operations applicable for two relations that are union-compatible
• UNION
• INTERSECTION
• SET DIFFERENCE or MINUS
– Set operations applicable to two relations that are not union-compatible
• CARTESIAN PRODUCT
10/09/2024
20 of 66

• Union Compatible (or Type Compatible relations)
– Two relations have the same no. of attributes & each corresponding pair of attributes
has the same domain
– i.e. Two relations R(A1, A2, ..., An) and S(B1, B2, ..., Bn) are said to be union compatible if
• (1) Both have same degree n
• (2) dom(Ai) = dom(Bi) for 1 ≤ i ≤ n
Two union-compatible relations
10/09/2024
21 of 66

• UNION (denoted as R S)
∪
– Result: A relation that includes all tuples that are either in R or S or both R and S
• INTERSECTION (denoted as R ∩ S)
– Result: A relation that includes all tuples that are in both R and S
• SET DIFFERENCE or MINUS (denoted by R – S)
– Result: A relation that includes all tuples that are in R but not in S
• Note:
– The resulting relation has the same attribute names as the first relation R
– Duplicate tuples are eliminated
– The attributes in the result can be renamed using the RENAME operator
10/09/2024
22 of 66

Two UNION compatible Relations
STUDENT ∪
INSTRUCTOR
STUDENT ∩ INSTRUCTOR
STUDENT – INSTRUCTOR
INSTRUCTOR – STUDENT
10/09/2024
23 of 67

Using UNION operation
10/09/2024
(To find the distinct SSN from EMPLOYEE with Dno = 5)
RESULT1 U RESULT2
DEP5_EMPS
24 of 67

• Note:
– Except for MINUS operations, UNION and INTERSECTION operations are commutative.
– Both UNION and INTERSECTION can be treated as n-ary operations (applicable to any
number of relations) because both are also associative operations
– INTERSECTION can be expressed in terms of UNION and DIFFERENCE
– In SQL,
• Set Operations (Eliminate duplicates): UNION, INTERSECT, and EXCEPT
• Multiset operations (Do not eliminate duplicates): UNION ALL, INTERSECT ALL, EXCEPT ALL
10/09/2024
25 of 66

• CARTESIAN PRODUCT (1/7)
– Also called CROSS PRODUCT or CROSS JOIN
– Denoted by x
• R(A1, A2, ..., An) × S(B1, B2, ..., Bm)
– If Q is the resulting relation
• Degree of Q: n + m
• Total tuples in Q
– If R has nR tuples (denoted as |R| = nR), and S has nS tuples, then Q will have nR * nS tuples
» i.e. Product of the number of rows in the two tables being joined
10/09/2024
26 of 66

– Example: Retrieve a list of names of each female employee’s dependents
10/09/2024
27 of 66

Using CARTESIAN PRODUCT operation
10/09/2024
(Retrieve a list of names of each female employee’s dependents)
28 of 67

10/09/2024
Using CARTESIAN PRODUCT operation
(Retrieve a list of names of each female employee’s dependents)
Order of Attributes No. of Tuples = 3x7 = 21
Degree = 3+5 = 8
29 of 67

Retrieve a list of names of each female
employee’s dependents
10/09/2024
30 of 67

– Useful when followed by a selection that matches values of attributes
• Introduces a new join operation to specify this sequence as a single operation
– In SQL, CARTESIAN PRODUCT can be realized by
• Using the CROSS JOIN option in joined tables
• Alternatively, If more than one relation is specified in FROM clause and there is no WHERE
clause, then CROSS PRODUCT of these relations are selected (except for duplicate
elimination)
10/09/2024
SELECT *
FROM Tab1
CROSS JOIN Tab2;
31 of 66

10/09/2024
– Every row from the first table is combined with every row from the second table,
resulting in a Cartesian product
– It does not require any join condition to be specified
– Can be computationally expensive and should be used with caution, especially when
joining large tables
SELECT *
FROM Tab1
CROSS JOIN Tab2;
32 of 66

Binary Relational Operations: JOIN (1/14)
• Denoted by
• Process relationships among relations
– General form of JOIN operation on two relations R(A1, A2, ..., An) & S(B1,B2, ..., Bm)
• <join condition> is of the form <condition> AND <condition> AND...AND <condition>
• If Q is the resulting relation
– Degree = Degree of Table-1 + Degree of Table-2
– Vs. Cartesian Product
• In JOIN, only combinations of tuples satisfying the join condition appear in the result
– i.e. One tuple in Q = Tuple of Table1 + Tuple of Table2 ONLY when the join condition is TRUE
– Tuples whose join attributes are NULL (or) for which join condition is FALSE do not appear in Q
• In CARTESIAN PRODUCT all combinations of tuples are included in the result
10/09/2024
33 of 66

• Example: Retrieve the name of the manager of each department
– (1) Combine each DEPARTMENT tuple with EMPLOYEE tuple if join condition is TRUE
– (2) PROJECT the result with suitable attributes
10/09/2024
34 of 66

10/09/2024
Retrieve the name of the manager of each department
PROJECT the result
with suitable
attributes
35 of 67

• JOIN can be specified as a CARTESIAN PRODUCT operation followed by a SELECT
operation (as discussed in Slide-32)
10/09/2024
36 of 66

10/09/2024
• NATURAL JOIN (1/4)
– Denoted by *
– <join condition>
• Automatically determined based on the columns with the same name in both tables
– If the names of join attributes are different, a RENAME operation is applied first
• It combines rows from both tables where the values in the matching columns are equal
– Removes duplicate columns from the result
– In SQL
SELECT *
FROM Tab1
NATURAL JOIN Tab2;
37 of 66

10/09/2024
38 of 67

• NATURAL JOIN (3/4)
– Example: Combine PROJECT with DEPARTMENT
• (1) The join attribute is Department number. As both tables should have same name of join
attribute, “Dnumber” attribute of DEPARTMENT should be renamed to “Dnum”
• (2) Apply NATURAL JOIN
10/09/2024
39 of 66

10/09/2024
Dnum
40 of 67

• EQUI JOIN (1/2)
– Denoted by
– EQUI JOIN are specific instances of THETA JOIN where <join condition> is explicitly
specified using the equality operator (=) on join attributes
– EQUI JOIN provides more control over <join condition> compared to NATURAL JOIN
– Tables are joined based on the equality of values in specified join attributes/columns
• Always have one or more pairs of attributes that have identical values in every tuple
– Most common type of join and is often used to match related rows between tables
– In SQL,
10/09/2024
SELECT *
FROM Tab1 A
JOIN Tab2 B ON A.id = B.id;
41 of 66

10/09/2024
• EQUI JOIN (2/2)
– Example:
Employee_ID Name Department_ID
1 John Doe 1
2 Jane Smith 1
3 Mike Johnson 2
4 Emily Brown 3
department_i
d department_name
1 Engineering
2 Marketing
3 Human Resources
DEPARTMENT
EMPLOYEE
EMPLOYEE_NAME DEPARTMENT_NAME
John Doe Engineering
Jane Smith Engineering
Mike Johnson Marketing
Emily Brown Human Resources
SELECT E.name AS employee_name, D.department_name
FROM Employees E
JOIN Departments D ON E.department_id = D.department_id
RESULTING RELATION
42 of 66

• THETA JOIN (1/2)
– Denoted by θ
– <join condition> is based on a general comparison condition (theta condition), which
can include any comparison operator such as '=', '>', '<', '>=', '<=', or '<>'
– THETA JOIN provides more flexibility than EQUI JOIN as they allow for <join condition>
other than equality
• i.e. Can be used to perform joins based on custom conditions that are not limited to
column equality
– Tuples whose join attributes are NULL (or) for which the join condition is FALSE do not
appear in the result
– In SQL
10/09/2024
SELECT *
FROM Tab1 A
JOIN Tab2 B ON A.col1 > B.col2;
43 of 66

10/09/2024
• THETA JOIN (2/2)
– Example
employee_id name department
1 John Doe Engineering
2 Jane Smith Engineering
3 Mike Johnson Marketing
4 Emily Brown Human Resources
employee_id salary
1 60000.00
2 65000.00
3 55000.00
4 60000.00
EMPLOYEES SALARIES
NAME
John Doe
Jane Smith
Emily Brown
SELECT Employees.name
FROM Employees
JOIN Salaries ON Employees.employee_id = Salaries.employee_id
WHERE Salaries.salary > 59000;
44 of 66

• Variations of JOIN
– n-way JOIN
• NATURAL JOIN or EQUIJOIN operation specified among multiple tables
– INNER JOIN
• Defined formally as a combination of CARTESIAN PRODUCT and SELECTION
• It is a broader concept that encompasses various types of join operations where only
matching rows from both tables are included in the result set
• In an INNER JOIN, rows from two tables are combined based on a specified condition,
which could be an equality condition (EQUI JOIN) or any other condition
• In SQL
10/09/2024
SELECT *
FROM Tab1 A
INNER JOIN Tab2 B ON A.id = B.id; -- Here, it is like EQUI JOIN
45 of 66

• In SQL, JOIN can be realized in several different ways
– (1) Specify <join conditions> in WHERE clause, along with other selection conditions
– (2) Use a nested relation
– (3) Use the concept of joined tables
• The construct of joined tables allows user to specify explicitly all the various types of join
• It also allows user to distinguish join conditions from selection conditions in WHERE clause
• Note:
– Join selectivity = Expected size of join result / maximum size
10/09/2024
46 of 66

Binary Relational Operations: DIVISION (1/5)
• Denoted by ÷
• Example-1:
– R ÷ S
• For a tuple(t) to appear in the resulting
relation (T), values in t must appear in R
in combination with every tuple in S
• Example-2:
– Retrieve names of EMPLOYEES who work
on all the projects that ‘John Smith’
WORKS ON
10/09/2024
a
a
a
a
a
a
47 of 66

SMITH
10/09/2024
48 of 67

10/09/2024
Retrieve names of EMPLOYEES who work on all the
projects that ‘John Smith’ WORKS ON
a
a
a
a
a
49 of 67

10/09/2024
Retrieve names of EMPLOYEES who work on all the projects that ‘John Smith’ WORKS ON
Result
50 of 67

Binary Relational Operations: DIVISION (5/5)
• DIVISION operation can be expressed as a sequence of π, ×, and – operations
10/09/2024
51 of 66

Operations of Relational Algebra (1/2)
10/09/2024
52 of 66

Operations of Relational Algebra (2/2)
10/09/2024
53 of 66

Query Tree (1/2)
• Data structure for internal representation of the Query in a RDBMS
• Also called as Query evaluation tree (or) Query execution tree
– Internal Representation
• Leaf Nodes => Input relations of query
• Internal Nodes => Relational algebra operations
• Query Execution
– Execution of an internal node operation starts when its operands (child nodes) are
available and it gets replaced by the relation that results from executing the operation
– Execution terminates when the root node is executed and produces the final result
relation for the query
10/09/2024
54 of 66

10/09/2024
55 of 67

Additional Relational Operations (1/10)
• Some operations cannot be specified in the basic original relational algebra
• Generalized projection
– Allows functions of attributes to be included in the projection list
10/09/2024
56 of 66

Generalized Projection (with renaming) to obtain a report from the relation EMPLOYEE
10/09/2024
57 of 67

• Aggregate functions and grouping
– Types
• (1) Mathematical Functions applied to collections of numeric values from database
– Example: SUM, AVERAGE, MAXIMUM, MINIMUM
• (2) Function applied to count tuples/values
– Example: COUNT
• (3) Group tuples by the value of some of their attributes and apply aggregate function
independently to each group
– Denoted by (pronounced script F)
– <grouping attributes> list of attributes of the relation specified in R
– <function list> list of (<function> <attribute>) pairs
» In each such pair <function> is one of the allowed functions (such as SUM, AVERAGE,
MAXIMUM, MINIMUM, COUNT), <attribute> is an attribute of relation R
10/09/2024
58 of 66

10/09/2024
59 of 67

Find the missing operation?
10/09/2024
60 of 67

• Recursive Closure operations
– Operation applied to a recursive relationship between tuples of same type
– SQL3 standard includes syntax for recursive closure
10/09/2024
61 of 66

• Outer Joins
– Keep all tuples in R, or all those in S, or all those in both relations regardless of
whether or not they have matching tuples in the other relation
– Types (part of the SQL2 standard)
• LEFT OUTER JOIN
• RIGHT OUTER JOIN
• FULL OUTER JOIN
– LEFT OUTER JOIN (denoted by )
• Keep every tuple in the first (or left) relation R in R S
• If NO matching tuple is found in S, then the attributes of S in the join result are filled or
padded with NULL values
10/09/2024
62 of 66

10/09/2024
63 of 67

• Outer Joins
– RIGHT OUTER JOIN (denoted by )
• Keep every tuple in the second (or right) relation S in R S
• If NO matching tuple is found in R, then the attributes of R in the join result are filled or
– FULL OUTER JOIN (denoted by )
• Keep every tuple in the left and right relation R and S
• If NO matching tuple is found in R and S, then their attributes in the join result are filled or
10/09/2024
64 of 66

• OUTER UNION Operation
– Union of tuples from two partially union (type) compatible relations R(X, Y) and S(X, Z)
that have some common attributes (X)
– Resulting Relation representation: T (X, Y, Z)
• Union compatible (X) attributes – represented ONLY once in the result
– Note: Same as FULL OUTER JOIN on the common attributes
• Non-Union compatible (Y, Z) attributes – Kept with a NULL value
10/09/2024
65 of 66

Thank You
10/09/2024
66 of 67

Relational Algebra in Database Management System

More Related Content

What's hot

Similar to Relational Algebra in Database Management System

More from Christalin Nelson

Recently uploaded

In this document

Relational Algebra in Database Management System