Relational_Algebra_Calculus Operations.pdf

Relational Algebra and Normalization
(Part – 2)
3/2/2024
Christalin Nelson | Systems | SoCS
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At a Glance
• Unary Relational Operations: SELECT and PROJECT
• Relational Algebra Operations from Set Theory
• Binary Relational Operations: JOIN and DIVISION
• Additional Relational Operations
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Relational Data Model
• Legacy DB Systems (1960s)
– Hierarchical and network models
• Relational model
– Introduced by Ted Codd (IBM, 1970)
– First commercial implementations (early 1980s)
• SQL/DS system on MVS OS by IBM, Oracle DBMS
– Has been implemented in a large number of commercial systems
– Formal languages used to query and manipulate data in relational DBs
• Relational algebra
• Relational calculus
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Relational Algebra and Relational Calculus (1/4)
• Relational algebra
– Procedural in nature
– Focuses on describing a sequence of operations to retrieve or manipulate data
– It provides a set of operations (such as SELECT, PROJECT, JOIN, UNION, INTERSECTION,
and DIFFERENCE) that act on relations (tables) to produce new relations as results
– Relational algebra operations can be thought of as similar to SQL query clauses (e.g.,
SELECT, FROM, WHERE, JOIN) but 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
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• Relational algebra operations can be divided into two groups
– 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)
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• Relational calculus
– Formal language based on the branch of mathematical logic called predicate calculus
– Declarative in nature
– Focuses on describing what data should be retrieved rather than how it should be
retrieved
• Relational calculus queries specify the conditions that the desired result must satisfy, and
the Database system is responsible for determining how to retrieve the data that meets
those conditions
– It provides a formal notation for specifying queries in terms of predicate logic & set
theory, without specifying the sequence of operations to be performed
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• Types of Relational Calculus
– Tuple relational calculus
• Specifies queries in terms of tuples (rows) that satisfy certain conditions
• Variables range over tuples
– Domain relational calculus
• Specifies queries in terms of domains (attribute values) that satisfy certain conditions
• Variables range over the domains (values) of attributes
• Query-By-Example (QBE) language
– A graphical user-friendly relational language based on domain relational calculus
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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>
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• 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|
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• 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
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• 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
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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
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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
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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
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Using intermediate relations and renaming of attributes
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• Rename attributes in intermediate results
• RENAME operation
– Denoted as rho
– Types
• Rename both relation & attributes =>
• Rename the relation only =>
• Rename the attributes only =>
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• 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
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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
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• 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
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• 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
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Two UNION compatible Relations
STUDENT ∪ INSTRUCTOR
STUDENT ∩ INSTRUCTOR
STUDENT – INSTRUCTOR
INSTRUCTOR – STUDENT
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Using UNION operation
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(To find the distinct SSN from EMPLOYEE with Dno = 5)
RESULT1 U RESULT2
DEP5_EMPS
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• 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
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• 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
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– Example: Retrieve a list of names of each female employee’s dependents
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Using CARTESIAN PRODUCT operation
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(Retrieve a list of names of each female employee’s dependents)
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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
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Retrieve a list of names of each female
employee’s dependents
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– 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)
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SELECT *
FROM Tab1
CROSS JOIN Tab2;
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– 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;
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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
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• 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
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Retrieve the name of the manager of each department
PROJECT the result
with suitable
attributes
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• JOIN can be specified as a CARTESIAN PRODUCT operation followed by a SELECT
operation (as discussed in Slide-32)
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• 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;
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• 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
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Dnum
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• 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,
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SELECT *
FROM Tab1 A
JOIN Tab2 B ON A.id = B.id;
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• 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_id 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
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• 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
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SELECT *
FROM Tab1 A
JOIN Tab2 B ON A.col1 > B.col2;
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• 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
SELECT Employees.name
FROM Employees
JOIN Salaries ON Employees.employee_id = Salaries.employee_id
WHERE Salaries.salary > 59000;
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• 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
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SELECT *
FROM Tab1 A
INNER JOIN Tab2 B ON A.id = B.id; -- Here, it is like EQUI JOIN
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• 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
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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
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a
a
a
a
a
a
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SMITH
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Retrieve names of EMPLOYEES who work on all the
projects that ‘John Smith’ WORKS ON
a
a
a
a
a
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Retrieve names of EMPLOYEES who work on all the projects that ‘John Smith’ WORKS ON
Result
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Binary Relational Operations: DIVISION (5/5)
• DIVISION operation can be expressed as a sequence of π, ×, and – operations
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Operations of Relational Algebra (1/2)
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Operations of Relational Algebra (2/2)
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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
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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
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Generalized Projection (with renaming) to obtain a report from the relation EMPLOYEE
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• 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
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Find the missing operation?
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• Recursive Closure operations
– Operation applied to a recursive relationship between tuples of same type
– SQL3 standard includes syntax for recursive closure
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• 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
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• 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
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• 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: T (X, Y, Z)
• The attributes that are union compatible (X) are represented only once in the result
– Note: Same as FULL OUTER JOIN on the common attributes
• The attributes that are not union compatible (Y, Z) from either relation are also kept with a
NULL value
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Thank You
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Relational_Algebra_Calculus Operations.pdf

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