This document provides an overview and agenda for a session on relational algebra, relational calculus, and SQL. It discusses the key topics of relational algebra operations like selection, projection, join, and division. It also covers the tuple relational calculus and domain relational calculus for specifying relational queries. Sample queries are provided in both relational algebra and relational calculus. The session aims to explain how relational algebra forms the basis for the SQL language for querying relational databases.
Unit-II DBMS presentation for students.pdfajajkhan16
Relational databases organize data into one or more tables made up of rows and columns to show relationships between different data structures. Relationships are logical connections between tables established based on interactions among the tables. Relational algebra provides theoretical foundations for relational databases and SQL through operators like select, project, join, and union that take relations as input and output new relations. Relational calculus is a non-procedural query language that uses predicates and quantifiers to specify what to retrieve from relations rather than how to retrieve it.
This document discusses relational algebra, which is a mathematical system used to represent and manipulate data in relational databases. It defines key concepts like base relations, which are stored in the database, and views, which are derived from base relations. The core operations of relational algebra are also summarized, including selection, projection, set operations, renaming, products, and various types of joins. Examples are provided to illustrate how each operation works on relations.
The document discusses relational algebra and relational calculus. It describes unary and binary relational operations in relational algebra such as select, project, union, intersection, difference. It also covers additional operations like join, division and aggregate functions. The document then discusses tuple relational calculus and domain relational calculus, explaining expressions, quantifiers, and how they are used to form relational queries.
The document discusses relational database query languages. Relational algebra is a procedural query language that takes relations as input and returns relations as output. It uses operators like select, project, rename, union, intersection, difference, and cartesian product. Query languages also support join operations like inner joins, theta joins, equi joins, natural joins, and outer joins like left, right, and full outer joins.
This document discusses the relational model and relational database concepts. It covers domains and relations, relational keys like primary keys, candidate keys, foreign keys and their rules. It also discusses relational operators, relational algebra, relational calculus, and the SQL language. Key types like alternate keys, candidate keys, compound keys, primary keys, superkeys, and foreign keys are defined. Relational algebra operations like selection, projection, renaming, union, intersection, difference, cartesian product, and join are explained. Tuple relational calculus and domain relational calculus are introduced. Examples of queries using relational algebra and calculus are provided. Components of SQL like DDL, DML, DCL are listed
This document discusses different concepts related to the relational model including domains and relations, relational data integrity, keys such as primary keys, candidate keys, foreign keys and their rules. It also discusses relational operators, relational algebra, relational calculus and SQL. Finally, it describes different types of relational algebra operations including unary operations like select, project and rename and binary operations like join, union, intersection, difference and cartesian product.
This document discusses the relational model and relational database concepts. It covers domains and relations, relational keys like primary keys, foreign keys, and candidate keys. It also discusses relational algebra operations like selection, projection, join, and set operations. Relational calculus is introduced. The SQL language components of DDL, DML, and DCL are mentioned for data definition, manipulation, and control. Key concepts like views, nested tables, and correlated subqueries are also summarized briefly.
Relational algebra is a set of operations used to specify queries on relational databases. The basic operations include select, project, join, union, intersection, and cartesian product. A sequence of relational algebra operations forms a query that manipulates relations and returns a relation of results. Common uses of the operations include retrieving subsets of tuples, projecting columns, joining tables, and set operations to combine relations.
Unit-II DBMS presentation for students.pdfajajkhan16
Relational databases organize data into one or more tables made up of rows and columns to show relationships between different data structures. Relationships are logical connections between tables established based on interactions among the tables. Relational algebra provides theoretical foundations for relational databases and SQL through operators like select, project, join, and union that take relations as input and output new relations. Relational calculus is a non-procedural query language that uses predicates and quantifiers to specify what to retrieve from relations rather than how to retrieve it.
This document discusses relational algebra, which is a mathematical system used to represent and manipulate data in relational databases. It defines key concepts like base relations, which are stored in the database, and views, which are derived from base relations. The core operations of relational algebra are also summarized, including selection, projection, set operations, renaming, products, and various types of joins. Examples are provided to illustrate how each operation works on relations.
The document discusses relational algebra and relational calculus. It describes unary and binary relational operations in relational algebra such as select, project, union, intersection, difference. It also covers additional operations like join, division and aggregate functions. The document then discusses tuple relational calculus and domain relational calculus, explaining expressions, quantifiers, and how they are used to form relational queries.
The document discusses relational database query languages. Relational algebra is a procedural query language that takes relations as input and returns relations as output. It uses operators like select, project, rename, union, intersection, difference, and cartesian product. Query languages also support join operations like inner joins, theta joins, equi joins, natural joins, and outer joins like left, right, and full outer joins.
This document discusses the relational model and relational database concepts. It covers domains and relations, relational keys like primary keys, candidate keys, foreign keys and their rules. It also discusses relational operators, relational algebra, relational calculus, and the SQL language. Key types like alternate keys, candidate keys, compound keys, primary keys, superkeys, and foreign keys are defined. Relational algebra operations like selection, projection, renaming, union, intersection, difference, cartesian product, and join are explained. Tuple relational calculus and domain relational calculus are introduced. Examples of queries using relational algebra and calculus are provided. Components of SQL like DDL, DML, DCL are listed
This document discusses different concepts related to the relational model including domains and relations, relational data integrity, keys such as primary keys, candidate keys, foreign keys and their rules. It also discusses relational operators, relational algebra, relational calculus and SQL. Finally, it describes different types of relational algebra operations including unary operations like select, project and rename and binary operations like join, union, intersection, difference and cartesian product.
This document discusses the relational model and relational database concepts. It covers domains and relations, relational keys like primary keys, foreign keys, and candidate keys. It also discusses relational algebra operations like selection, projection, join, and set operations. Relational calculus is introduced. The SQL language components of DDL, DML, and DCL are mentioned for data definition, manipulation, and control. Key concepts like views, nested tables, and correlated subqueries are also summarized briefly.
Relational algebra is a set of operations used to specify queries on relational databases. The basic operations include select, project, join, union, intersection, and cartesian product. A sequence of relational algebra operations forms a query that manipulates relations and returns a relation of results. Common uses of the operations include retrieving subsets of tuples, projecting columns, joining tables, and set operations to combine relations.
Query Decomposition and data localization Hafiz faiz
This document discusses query processing in distributed databases. It describes query decomposition, which transforms a high-level query into an equivalent lower-level algebraic query. The main steps in query decomposition are normalization, analysis, redundancy elimination, and rewriting the query in relational algebra. Data localization then translates the algebraic query on global relations into a query on physical database fragments using fragmentation rules.
Relational algebra is the formal system used to manipulate data in a relational database. It consists of operators like select, project, join, union, and difference that are applied to relations to retrieve or modify data. Relational calculus provides an alternative, non-procedural way to query databases using logic and variables. Both are based on relational and algebraic concepts and underpin the functionality of relational databases and SQL.
Relational algebra is a theoretical procedural language used to define and manipulate relations through a set of operations. The basic relational algebra operations are selection, projection, Cartesian product, union, and set difference. Other operations like join, intersection, and theta join can be expressed in terms of these basic operations. Relational algebra allows queries to be formed and nested through the application of multiple operations on relations.
The document discusses relational algebra and operations that can be performed on relations in a relational database. It describes unary operations like select and project that operate on a single relation, as well as binary operations like join, union, and difference that combine two relations. It also covers additional operations like aggregation, outer joins, and recursive closure that provide more advanced querying capabilities beyond the core relational algebra. The goal of relational algebra is to allow users to use a sequence of algebraic operations to specify database queries and retrieve relation data in a declarative manner.
The document discusses relational algebra and normalization. It describes unary operations like SELECT and PROJECT that act on single relations. It also describes binary operations from set theory like UNION, INTERSECTION, and CARTESIAN PRODUCT that combine two relations. JOIN is introduced as a binary operation that combines relations based on a join condition. NATURAL JOIN is a specific type of JOIN that automatically matches columns with the same name between relations.
Introduction to Relational Database Management SystemsAdri Jovin
This presentation contains information relevant to the relational database management systems and the various operations that are performed over a relational database.
This document provides an overview of relational algebra and calculus, database normalization, and queries. It defines common relational algebra operations like selection, projection, join, etc. It explains database normalization forms like 1NF, 2NF, 3NF and their advantages. It also covers functional dependencies, integrity constraints, and different types of queries including subqueries and nested subqueries.
This document provides an overview of relational algebra operations. There are five basic relational algebra operators: select, project, union, intersection, and cartesian product. The select and project operators are unary operators that operate on a single relation. Select filters rows based on a predicate, while project extracts specified column values. Union and intersection are binary operators that combine two relations, union returning all rows and intersection returning matching rows. Cartesian product returns all possible combinations of rows from two relations. Relational algebra provides a way to formulate queries using these relational operators.
Here are the relational algebra and SQL queries for the statements:
1. Find names of sailors who’ve reserved boat #103
[RELational Algebra]
πsname(σbid=103(Sailors ⋈ Reserves))
[SQL]
SELECT sname FROM Sailors, Reserves WHERE sid = sid AND bid = 103
2. Find names of sailors who’ve reserved a red boat
[RELational Algebra]
πsname(σcolor='Red'(Boats ⋈ Reserves ⋈ Sailors)))
[SQL]
SELECT sname FROM Sailors, Reserves, Boats WHERE sid = sid AND bid =
This document provides an overview of relational algebra operations. There are five basic relational algebra operators: select, project, union, intersection, and cartesian product. Select retrieves rows from a table that meet a predicate condition. Project extracts values of specified columns from a table. Union combines the rows from two tables, intersection returns rows that exist in both tables, and cartesian product returns all possible combinations of rows from two tables. Relational algebra operations can be composed to form more complex queries.
1. Query optimization involves transforming relational expressions using equivalence rules to generate alternative query plans and choosing the cheapest plan based on estimated cost.
2. Equivalence rules allow expressions to be rewritten while maintaining logical equivalence, such as pushing down selections early in the query execution to reduce the size of intermediate results.
3. Join ordering and application of projections can further optimize query execution by minimizing the size of temporary relations that need to be stored and processed.
This slide deck is used as an introduction to Relational Algebra and its relation to the MapReduce programming model, as part of the Distributed Systems and Cloud Computing course I hold at Eurecom.
Course website:
http://michiard.github.io/DISC-CLOUD-COURSE/
Sources available here:
https://github.com/michiard/DISC-CLOUD-COURSE
This document discusses relational algebra operations used to manipulate relations in relational databases. It covers selection, projection, set operations including union, intersection, difference, and joins. Selection chooses tuples that satisfy a condition, projection extracts specified attributes. Union combines relations, intersection returns common elements, difference returns elements in one relation not in another. Joins combine rows from tables based on a related column.
Here, we talk about various relational algebra operations like select, project, union, intersection, minus, cartesian product, and join in database management systems.
Relational algebra and calculus are formal query languages used to manipulate and retrieve data from relational databases. Relational algebra uses algebraic operations like selection, projection, join, etc. to represent queries procedurally. Relational calculus allows users to describe what data is wanted declaratively using logic-based formulas with variables, quantifiers and predicates over relation instances. Both have free and bound variables, and queries return tuples that satisfy the formulas by assigning constants to free variables.
The document discusses Taguchi techniques for designing robust and quality products. Some key points:
- Taguchi advocated designing quality into products through robust design using orthogonal arrays rather than inspecting quality afterwards.
- Orthogonal arrays allow experiments to be conducted efficiently with many factors and interactions studied using few experimental runs.
- Steps involve identifying factors, assigning them to orthogonal array columns, conducting the experiment, analyzing results and selecting optimal levels.
Relational algebra is the basic set of operations for the relational model, including unary operations like selection and projection, binary operations like various join types, and set operations like union and intersection. Relational algebra operations manipulate relations and produce new relations, allowing users to specify database queries. Common operations include selection to filter tuples, projection to select attributes, equijoins to match tuples on equality conditions, and outer joins to retain non-matching tuples.
This document summarizes key concepts from Chapter 2 on the relational database model. It introduces relations as tables with rows and columns, and discusses attribute types, relation schemas, and relation instances. It covers the concepts of keys and foreign keys. It also summarizes the core relational algebra operations of selection, projection, union, intersection, difference, Cartesian product, natural join, and renaming. Finally, it notes that relational algebra is not Turing complete and cannot perform aggregation operations like SUM and AVG.
The document discusses the relational algebra and its fundamental operations. It describes relational algebra as the basic set of operations for the relational model that enable users to specify basic retrieval requests. The fundamental operations include unary operations like select and project, and binary operations like union, intersection, set difference, cartesian product, join, and outer join. Each operation is explained along with examples of how it is denoted and used.
Software Engineering and Project Management - Software Testing + Agile Method...Prakhyath Rai
Software Testing: A Strategic Approach to Software Testing, Strategic Issues, Test Strategies for Conventional Software, Test Strategies for Object -Oriented Software, Validation Testing, System Testing, The Art of Debugging.
Agile Methodology: Before Agile – Waterfall, Agile Development.
Query Decomposition and data localization Hafiz faiz
This document discusses query processing in distributed databases. It describes query decomposition, which transforms a high-level query into an equivalent lower-level algebraic query. The main steps in query decomposition are normalization, analysis, redundancy elimination, and rewriting the query in relational algebra. Data localization then translates the algebraic query on global relations into a query on physical database fragments using fragmentation rules.
Relational algebra is the formal system used to manipulate data in a relational database. It consists of operators like select, project, join, union, and difference that are applied to relations to retrieve or modify data. Relational calculus provides an alternative, non-procedural way to query databases using logic and variables. Both are based on relational and algebraic concepts and underpin the functionality of relational databases and SQL.
Relational algebra is a theoretical procedural language used to define and manipulate relations through a set of operations. The basic relational algebra operations are selection, projection, Cartesian product, union, and set difference. Other operations like join, intersection, and theta join can be expressed in terms of these basic operations. Relational algebra allows queries to be formed and nested through the application of multiple operations on relations.
The document discusses relational algebra and operations that can be performed on relations in a relational database. It describes unary operations like select and project that operate on a single relation, as well as binary operations like join, union, and difference that combine two relations. It also covers additional operations like aggregation, outer joins, and recursive closure that provide more advanced querying capabilities beyond the core relational algebra. The goal of relational algebra is to allow users to use a sequence of algebraic operations to specify database queries and retrieve relation data in a declarative manner.
The document discusses relational algebra and normalization. It describes unary operations like SELECT and PROJECT that act on single relations. It also describes binary operations from set theory like UNION, INTERSECTION, and CARTESIAN PRODUCT that combine two relations. JOIN is introduced as a binary operation that combines relations based on a join condition. NATURAL JOIN is a specific type of JOIN that automatically matches columns with the same name between relations.
Introduction to Relational Database Management SystemsAdri Jovin
This presentation contains information relevant to the relational database management systems and the various operations that are performed over a relational database.
This document provides an overview of relational algebra and calculus, database normalization, and queries. It defines common relational algebra operations like selection, projection, join, etc. It explains database normalization forms like 1NF, 2NF, 3NF and their advantages. It also covers functional dependencies, integrity constraints, and different types of queries including subqueries and nested subqueries.
This document provides an overview of relational algebra operations. There are five basic relational algebra operators: select, project, union, intersection, and cartesian product. The select and project operators are unary operators that operate on a single relation. Select filters rows based on a predicate, while project extracts specified column values. Union and intersection are binary operators that combine two relations, union returning all rows and intersection returning matching rows. Cartesian product returns all possible combinations of rows from two relations. Relational algebra provides a way to formulate queries using these relational operators.
Here are the relational algebra and SQL queries for the statements:
1. Find names of sailors who’ve reserved boat #103
[RELational Algebra]
πsname(σbid=103(Sailors ⋈ Reserves))
[SQL]
SELECT sname FROM Sailors, Reserves WHERE sid = sid AND bid = 103
2. Find names of sailors who’ve reserved a red boat
[RELational Algebra]
πsname(σcolor='Red'(Boats ⋈ Reserves ⋈ Sailors)))
[SQL]
SELECT sname FROM Sailors, Reserves, Boats WHERE sid = sid AND bid =
This document provides an overview of relational algebra operations. There are five basic relational algebra operators: select, project, union, intersection, and cartesian product. Select retrieves rows from a table that meet a predicate condition. Project extracts values of specified columns from a table. Union combines the rows from two tables, intersection returns rows that exist in both tables, and cartesian product returns all possible combinations of rows from two tables. Relational algebra operations can be composed to form more complex queries.
1. Query optimization involves transforming relational expressions using equivalence rules to generate alternative query plans and choosing the cheapest plan based on estimated cost.
2. Equivalence rules allow expressions to be rewritten while maintaining logical equivalence, such as pushing down selections early in the query execution to reduce the size of intermediate results.
3. Join ordering and application of projections can further optimize query execution by minimizing the size of temporary relations that need to be stored and processed.
This slide deck is used as an introduction to Relational Algebra and its relation to the MapReduce programming model, as part of the Distributed Systems and Cloud Computing course I hold at Eurecom.
Course website:
http://michiard.github.io/DISC-CLOUD-COURSE/
Sources available here:
https://github.com/michiard/DISC-CLOUD-COURSE
This document discusses relational algebra operations used to manipulate relations in relational databases. It covers selection, projection, set operations including union, intersection, difference, and joins. Selection chooses tuples that satisfy a condition, projection extracts specified attributes. Union combines relations, intersection returns common elements, difference returns elements in one relation not in another. Joins combine rows from tables based on a related column.
Here, we talk about various relational algebra operations like select, project, union, intersection, minus, cartesian product, and join in database management systems.
Relational algebra and calculus are formal query languages used to manipulate and retrieve data from relational databases. Relational algebra uses algebraic operations like selection, projection, join, etc. to represent queries procedurally. Relational calculus allows users to describe what data is wanted declaratively using logic-based formulas with variables, quantifiers and predicates over relation instances. Both have free and bound variables, and queries return tuples that satisfy the formulas by assigning constants to free variables.
The document discusses Taguchi techniques for designing robust and quality products. Some key points:
- Taguchi advocated designing quality into products through robust design using orthogonal arrays rather than inspecting quality afterwards.
- Orthogonal arrays allow experiments to be conducted efficiently with many factors and interactions studied using few experimental runs.
- Steps involve identifying factors, assigning them to orthogonal array columns, conducting the experiment, analyzing results and selecting optimal levels.
Relational algebra is the basic set of operations for the relational model, including unary operations like selection and projection, binary operations like various join types, and set operations like union and intersection. Relational algebra operations manipulate relations and produce new relations, allowing users to specify database queries. Common operations include selection to filter tuples, projection to select attributes, equijoins to match tuples on equality conditions, and outer joins to retain non-matching tuples.
This document summarizes key concepts from Chapter 2 on the relational database model. It introduces relations as tables with rows and columns, and discusses attribute types, relation schemas, and relation instances. It covers the concepts of keys and foreign keys. It also summarizes the core relational algebra operations of selection, projection, union, intersection, difference, Cartesian product, natural join, and renaming. Finally, it notes that relational algebra is not Turing complete and cannot perform aggregation operations like SUM and AVG.
The document discusses the relational algebra and its fundamental operations. It describes relational algebra as the basic set of operations for the relational model that enable users to specify basic retrieval requests. The fundamental operations include unary operations like select and project, and binary operations like union, intersection, set difference, cartesian product, join, and outer join. Each operation is explained along with examples of how it is denoted and used.
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Software Testing: A Strategic Approach to Software Testing, Strategic Issues, Test Strategies for Conventional Software, Test Strategies for Object -Oriented Software, Validation Testing, System Testing, The Art of Debugging.
Agile Methodology: Before Agile – Waterfall, Agile Development.
Use PyCharm for remote debugging of WSL on a Windo cf5c162d672e4e58b4dde5d797...shadow0702a
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Digital Twins Computer Networking Paper Presentation.pptxaryanpankaj78
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DEEP LEARNING FOR SMART GRID INTRUSION DETECTION: A HYBRID CNN-LSTM-BASED MODELijaia
As digital technology becomes more deeply embedded in power systems, protecting the communication
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Robust Intrusion Detection Systems (IDS) are necessary for early threat detection and mitigation because
of the interconnection of these networks, which makes them vulnerable to a variety of cyberattacks. To
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dataset (DNP3), which focuses on unauthorized commands and Denial of Service (DoS) cyberattacks, to
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Null Bangalore | Pentesters Approach to AWS IAMDivyanshu
#Abstract:
- Learn more about the real-world methods for auditing AWS IAM (Identity and Access Management) as a pentester. So let us proceed with a brief discussion of IAM as well as some typical misconfigurations and their potential exploits in order to reinforce the understanding of IAM security best practices.
- Gain actionable insights into AWS IAM policies and roles, using hands on approach.
#Prerequisites:
- Basic understanding of AWS services and architecture
- Familiarity with cloud security concepts
- Experience using the AWS Management Console or AWS CLI.
- For hands on lab create account on [killercoda.com](https://killercoda.com/cloudsecurity-scenario/)
# Scenario Covered:
- Basics of IAM in AWS
- Implementing IAM Policies with Least Privilege to Manage S3 Bucket
- Objective: Create an S3 bucket with least privilege IAM policy and validate access.
- Steps:
- Create S3 bucket.
- Attach least privilege policy to IAM user.
- Validate access.
- Exploiting IAM PassRole Misconfiguration
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- Objective: Demonstrate how a PassRole misconfiguration can grant unauthorized access.
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- Allow user to pass IAM role to EC2.
- Exploit misconfiguration for unauthorized access.
- Access sensitive resources.
- Exploiting IAM AssumeRole Misconfiguration with Overly Permissive Role
- An overly permissive IAM role configuration can lead to privilege escalation by creating a role with administrative privileges and allow a user to assume this role.
- Objective: Show how overly permissive IAM roles can lead to privilege escalation.
- Steps:
- Create role with administrative privileges.
- Allow user to assume the role.
- Perform administrative actions.
- Differentiation between PassRole vs AssumeRole
Try at [killercoda.com](https://killercoda.com/cloudsecurity-scenario/)
Electric vehicle and photovoltaic advanced roles in enhancing the financial p...IJECEIAES
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4. 4
What is the class about?
Course description and syllabus:
» http://www.nyu.edu/classes/jcf/CSCI-GA.2433-001
» http://cs.nyu.edu/courses/fall11/CSCI-GA.2433-001/
Textbooks:
» Fundamentals of Database Systems (6th Edition)
Ramez Elmasri and Shamkant Navathe
Addition Wesley
ISBN-10: 0-1360-8620-9, ISBN-13: 978-0136086208 6th Edition (04/10)
5. 5
Icons / Metaphors
5
Common Realization
Information
Knowledge/Competency Pattern
Governance
Alignment
Solution Approach
6. 6
Agenda
1 Session Overview
5 Summary and Conclusion
2 Relational Algebra and Relational Calculus
3 Relational Algebra Using SQL Syntax
7. 7
Agenda
Unary Relational Operations: SELECT and
PROJECT
Relational Algebra Operations from Set Theory
Binary Relational Operations: JOIN and
DIVISION
Additional Relational Operations
Examples of Queries in Relational Algebra
The Tuple Relational Calculus
The Domain Relational Calculus
8. 8
The Relational Algebra and Relational Calculus
Relational algebra
Basic set of operations for the relational model
Relational algebra expression
Sequence of relational algebra operations
Relational calculus
Higher-level declarative language for specifying
relational queries
9. 9
Unary Relational Operations: SELECT and PROJECT (1/3)
The SELECT Operation
Subset of the tuples from a relation that
satisfies a selection condition:
• Boolean expression contains clauses of the form
<attribute name> <comparison op> <constant
value>
• or
• <attribute name> <comparison op> <attribute
name>
10. 10
Unary Relational Operations: SELECT and PROJECT (2/3)
Example:
<selection condition> applied independently
to each individual tuple t in R
If condition evaluates to TRUE, tuple selected
Boolean conditions AND, OR, and NOT
Unary
Applied to a single relation
11. 11
Unary Relational Operations: SELECT and PROJECT (3/3)
Selectivity
Fraction of tuples selected by a selection
condition
SELECT operation commutative
Cascade SELECT operations into a single
operation with AND condition
12. 12
The PROJECT Operation
Selects columns from table and discards
the other columns:
Degree
Number of attributes in <attribute list>
Duplicate elimination
Result of PROJECT operation is a set of
distinct tuples
13. 13
Sequences of Operations and the RENAME Operation
In-line expression:
Sequence of operations:
Rename attributes in intermediate results
RENAME operation
14. 14
Relational Algebra Operations from Set Theory (1/2)
UNION, INTERSECTION, and MINUS
Merge the elements of two sets in various ways
Binary operations
Relations must have the same type of tuples
UNION
R U S
Includes all tuples that are either in R or in S or
in both R and S
Duplicate tuples eliminated
15. 15
Relational Algebra Operations from Set Theory (2/2)
INTERSECTION
R ∩ S
Includes all tuples that are in both R and S
SET DIFFERENCE (or MINUS)
R – S
Includes all tuples that are in R but not in S
16. 16
The CARTESIAN PRODUCT (CROSS PRODUCT) Operation
CARTESIAN PRODUCT
CROSS PRODUCT or CROSS JOIN
Denoted by ×
Binary set operation
Relations do not have to be union compatible
Useful when followed by a selection that
matches values of attributes
17. 17
Binary Relational Operations: JOIN and DIVISION (1/2)
The JOIN Operation
Denoted by
Combine related tuples from two relations into
single “longer” tuples
General join condition of the form <condition>
AND <condition> AND...AND <condition>
Example:
18. 18
Binary Relational Operations: JOIN and DIVISION (2/2)
THETA JOIN
Each <condition> of the form Ai θ Bj
Ai is an attribute of R
Bj is an attribute of S
Ai and Bj have the same domain
θ (theta) is one of the comparison operators:
• {=, <, ≤, >, ≥, ≠}
19. 19
Variations of JOIN: The EQUIJOIN and NATURAL JOIN (1/2)
EQUIJOIN
Only = comparison operator used
Always have one or more pairs of attributes
that have identical values in every tuple
NATURAL JOIN
Denoted by *
Removes second (superfluous) attribute in an
EQUIJOIN condition
20. 20
Variations of JOIN: The EQUIJOIN and NATURAL JOIN (2/2)
Join selectivity
Expected size of join result divided by the
maximum size nR * nS
Inner joins
Type of match and combine operation
Defined formally as a combination of
CARTESIAN PRODUCT and SELECTION
21. 21
A Complete Set of Relational Algebra Operations
Set of relational algebra operations {σ, π,
U, ρ, –, ×} is a complete set
Any relational algebra operation can be
expressed as a sequence of operations from
this set
22. 22
The DIVISION Operation
Denoted by ÷
Example: retrieve the names of employees
who work on all the projects that ‘John
Smith’ works on
Apply to relations R(Z) ÷ S(X)
Attributes of R are a subset of the attributes of
S
25. 25
Notation for Query Trees
Query tree
Represents the input relations of query as leaf
nodes of the tree
Represents the relational algebra operations
as internal nodes
27. 27
Additional Relational Operations (1/2)
Generalized projection
Allows functions of attributes to be included in
the projection list
Aggregate functions and grouping
Common functions applied to collections of
numeric values
Include SUM, AVERAGE, MAXIMUM, and
MINIMUM
28. 28
Additional Relational Operations (2/2)
Group tuples by the value of some of their
attributes
Apply aggregate function independently to
each group
31. 31
OUTER JOIN Operations
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
• LEFT OUTER JOIN, RIGHT OUTER JOIN, FULL
OUTER JOIN
Example:
32. 32
The OUTER UNION Operation
Take union of tuples from two relations that
have some common attributes
Not union (type) compatible
Partially compatible
All tuples from both relations included in the
result
Tuples with the same value combination will
appear only once
36. 36
The Tuple Relational Calculus
Declarative expression
Specify a retrieval request
Non-procedural language
Any retrieval that can be specified in basic
relational algebra
Can also be specified in relational calculus
37. 37
Tuple Variables and Range Relations
Tuple variables
Ranges over a particular database relation
Satisfy COND(t):
Specify:
Range relation R of t
Select particular combinations of tuples
Set of attributes to be retrieved (requested
attributes)
38. 38
Expressions and Formulas in Tuple Relational Calculus
General expression of tuple relational
calculus is of the form:
Truth value of an atom
Evaluates to either TRUE or FALSE for a
specific combination of tuples
Formula (Boolean condition)
Made up of one or more atoms connected via
logical operators AND, OR, and NOT
39. 39
Existential and Universal Quantifiers
Universal quantifier (inverted “A”)
Existential quantifier (mirrored “E”)
Define a tuple variable in a formula as free
or bound
42. 42
Transforming the Universal and Existential Quantifiers
Transform one type of quantifier into other
with negation (preceded by NOT)
AND and OR replace one another
Negated formula becomes un-negated
Un-negated formula becomes negated
44. 44
Safe Expressions
Guaranteed to yield a finite number of
tuples as its result
Otherwise expression is called unsafe
Expression is safe
If all values in its result are from the domain of
the expression
45. 45
The Domain Relational Calculus (1/2)
Differs from tuple calculus in type of
variables used in formulas
Variables range over single values from
domains of attributes
Formula is made up of atoms
Evaluate to either TRUE or FALSE for a
specific set of values
• Called the truth values of the atoms
47. 47
Summary
Formal languages for relational model of
data:
Relational algebra: operations, unary and
binary operators
Some queries cannot be stated with basic
relational algebra operations
• But are important for practical use
Relational calculus
Based predicate calculus
48. 48
Agenda
1 Session Overview
4 Summary and Conclusion
2 Relational Algebra and Relational Calculus
3 Relational Algebra Using SQL Syntax
49. 49
Agenda
Relational Algebra and SQL
Basic Syntax Comparison
Sets and Operations on Relations
Relations in Relational Algebra
Empty Relations
Relational Algebra vs. Full SQL
Operations on Relations
» Projection
» Selection
» Cartesian Product
» Union
» Difference
» Intersection
From Relational Algebra to Queries (with Examples)
Microsoft Access Case Study
Pure Relational Algebra
50. 50
Relational Algebra And SQL
SQL is based on relational algebra with many extensions
» Some necessary
» Some unnecessary
“Pure” relational algebra, use mathematical notation with Greek
letters
It is covered here using SQL syntax; that is this unit covers
relational algebra, but it looks like SQL
And will be really valid SQL
Pure relational algebra is used in research, scientific papers, and
some textbooks
So it is good to know it, and material is provided at the end of this
unit material from which one can learn it
But in anything practical, including commercial systems, you will be
using SQL
51. 51
Key Differences Between SQL And “Pure” Relational Algebra
SQL data model is a multiset not a set; still rows in tables (we
sometimes continue calling relations)
» Still no order among rows: no such thing as 1st row
» We can (if we want to) count how many times a particular row appears
in the table
» We can remove/not remove duplicates as we specify (most of the time)
» There are some operators that specifically pay attention to duplicates
» We must know whether duplicates are removed (and how) for each
SQL operation; luckily, easy
Many redundant operators (relational algebra had only one:
intersection)
SQL provides statistical operators, such as AVG (average)
» Can be performed on subsets of rows; e.g. average salary per
company branch
52. 52
Key Differences Between Relational Algebra And SQL
Every domain is “enhanced” with a special
element: NULL
» Very strange semantics for handling these
elements
“Pretty printing” of output: sorting, and
similar
Operations for
» Inserting
» Deleting
» Changing/updating (sometimes not easily
reducible to deleting and inserting)
53. 53
Basic Syntax Comparison (1/2)
Relational Algebra SQL
p a, b SELECT a, b
s (d > e) (f =g ) WHERE d > e AND f = g
p q FROM p, q
p a, b s (d > e) (f =g ) (p q) SELECT a, b
FROM p, q
WHERE d > e AND f = g;
{must always have SELECT even if all
attributes are kept, can be written as:
SELECT *}
renaming AS {or blank space}
p := result INSERT INTO p
result {assuming p was empty}
p a, b (p) (assume a, b are the only
attributes)
SELECT *
FROM p;
54. 54
Basic Syntax Comparison (2/2)
Relational Algebra SQL
p q SELECT *
FROM p
UNION
SELECT *
FROM q
p − q SELECT *
FROM p
EXCEPT
SELECT *
FROM q
Sometimes, instead, we have DELETE
FROM
p q SELECT *
FROM p
INTERSECT
SELECT *
FROM q
55. 55
Sets And Operations On Them
If A, B, and C are sets, then we have the operations
Union, A B = { x x A x B }
Intersection, A B = { x x A x B }
Difference, A B = { x x A x B }
Cartesian product, A B = { (x,y) x A y B }, A
B C = { (x,y,z) x A y B z B }, etc.
The above operations form an algebra, that is you can
perform operations on results of operations, such as (A
B) (C A)
So you can write expressions and not just programs!
56. 56
Relations in Relational Algebra
Relations are sets of tuples, which we will also call
rows, drawn from some domains
Relational algebra deals with relations (which look like
tables with fixed number of columns and varying
number of rows)
We assume that each domain is linearly ordered, so for
each x and y from the domain, one of the following
holds
» x < y
» x = y
» x < y
Frequently, such comparisons will be meaningful even if
x and y are drawn from different columns
» For example, one column deals with income and another with
expenditure: we may want to compare them
57. 57
Reminder: Relations in Relational Algebra
The order of rows and whether a row appears once or
many times does not matter
The order of columns matters, but as our columns will
always be labeled, we will be able to reconstruct the
order even if the columns are permuted.
The following two relations are equal:
R A B
1 10
2 20
R B A
20 2
10 1
20 2
20 2
58. 58
Many Empty Relations
In set theory, there is only one empty set
For us, it is more convenient to think that for each
relation schema, that for specific choice of column
names and domains, there is a different empty relation
And of, course, two empty relations with different
number of columns must be different
So for instance the two relations below are different
The above needs to be stated more precisely to be
“completely correct,” but as this will be intuitively clear,
we do not need to worry about this too much
59. 59
Relational Algebra Versus Full SQL
Relational algebra is restricted to querying the database
Does not have support for
» Primary keys
» Foreign keys
» Inserting data
» Deleting data
» Updating data
» Indexing
» Recovery
» Concurrency
» Security
» …
Does not care about efficiency, only about specifications
of what is needed
60. 60
Operations on relations
There are several fundamental operations on
relations
We will describe them in turn:
» Projection
» Selection
» Cartesian product
» Union
» Difference
» Intersection (technically not fundamental)
The very important property: Any operation on
relations produces a relation
This is why we call this an algebra
61. 61
Projection: Choice Of Columns
SQL statement
SELECT B, A, D
FROM R
We could have removed the duplicate
row, but did not have to
R A B C D
1 10 100 1000
1 20 100 1000
1 20 200 1000
B A D
10 1 1000
20 1 1000
20 1 1000
62. 62
Selection: Choice Of Rows
SQL statement:
SELECT * (this means all columns)
FROM R
WHERE A <= C AND D = 4; (this is a predicate, i.e., condition)
R A B C D
5 5 7 4
5 6 5 7
4 5 4 4
5 5 5 5
4 6 5 3
4 4 3 4
4 4 4 5
4 6 4 6
A B C D
5 5 7 4
4 5 4 4
63. 63
Selection
In general, the condition (predicate) can be
specified by a Boolean formula with
NOT, AND, OR on atomic conditions, where a
condition is:
» a comparison between two column names,
» a comparison between a column name and a
constant
» Technically, a constant should be put in quotes
» Even a number, such as 4, perhaps should be put in
quotes, as ‘4’, so that it is distinguished from a
column name, but as we will never use numbers for
column names, this not necessary
64. 64
Cartesian Product
SQL statement
SELECT A, R.B, C, S.B, D
FROM R, S; (comma stands for Cartesian product)
A R.B C S.B D
1 10 40 10 10
1 10 50 20 10
2 10 40 10 10
2 10 50 20 10
2 20 40 10 10
2 20 50 20 10
R A B
1 10
2 10
2 20
S C B D
40 10 10
50 20 10
65. 65
A Typical Use Of Cartesian Product
SQL statement:
SELECT ID#, R.Room#, Size
FROM R, S
WHERE R.Room# = S.Room#;
R Size Room#
140 1010
150 1020
140 1030
S ID# Room# YOB
40 1010 1982
50 1020 1985
ID# R.Room# Size
40 1010 140
50 1020 150
66. 66
A Typical Use Of Cartesian Product
After the Cartesian product, we got
This allowed us to correlate the information from
the two original tables by examining each tuple
in turn
Size R.Room# ID# S.Room
#
YOB
140 1010 40 1010 1982
140 1010 50 1020 1985
150 1020 40 1010 1982
150 1020 50 1020 1985
140 1030 40 1010 1982
140 1030 50 1020 1985
67. 67
A Typical Use Of Cartesian Product
This example showed how to correlate information from
two tables
» The first table had information about rooms and their sizes
» The second table had information about employees including
the rooms they sit in
» The resulting table allows us to find out what are the sizes of
the rooms the employees sit in
We had to specify R.Room# or S.Room#, even though
they happen to be equal due to the specific equality
condition
We could, as we will see later, rename a column, to get
Room#
ID# Room# Size
40 1010 140
50 1020 150
68. 68
Union
SQL statement
(SELECT *
FROM R)
UNION
(SELECT *
FROM S);
Note: We happened to choose to remove duplicate
rows
Note: we could not just write R UNION S (syntax quirk)
R A B
1 10
2 20
S A B
1 10
3 20
A B
1 10
2 20
3 20
69. 69
Union Compatibility
We require same -arity (number of columns),
otherwise the result is not a relation
Also, the operation “probably” should make
sense, that is the values in corresponding
columns should be drawn from the same
domains
Actually, best to assume that the column names
are the same and that is what we will do from
now on
We refer to these as union compatibility of
relations
Sometimes, just the term compatibility is used
70. 70
Difference
SQL statement
(SELECT *
FROM R)
MINUS
(SELECT *
FROM S);
Union compatibility required
EXCEPT is a synonym for MINUS
R A B
1 10
2 20
S A B
1 10
3 20
A B
2 20
71. 71
Intersection
SQL statement
(SELECT *
FROM R)
INTERSECT
(SELECT *
FROM S);
Union compatibility required
Can be computed using differences only: R – (R – S)
R A B
1 10
2 20
S A B
1 10
3 20
A B
1 10
72. 72
From Relational Algebra to Queries
These operations allow us to define a large
number of interesting queries for relational
databases.
In order to be able to formulate our examples,
we will assume standard programming
language type of operations:
» Assignment of an expression to a new variable;
In our case assignment of a relational expression to
a relational variable.
» Renaming of a relations, to use another name to
denote it
» Renaming of a column, to use another name to
denote it
73. 73
A Small Example
The example consists of 3 relations:
Person(Name,Sex,Age)
This relation, whose primary key is Name, gives information about the
human’s sex and age
Birth(Parent,Child)
This relation, whose primary key is the pair Parent,Child, with both being
foreign keys referring to Person gives information about who is a parent of
whom. (Both mother and father would be generally listed)
Marriage(Husband,Wife, Age) or
Marriage(Husband,Wife, Age)
This relation listing current marriages only, requires choosing which spouse
will serve as primary key. For our exercise, it does not matter what the
choice is. Both Husband and Wife are foreign keys referring to Person. Age
specifies how long the marriage has lasted.
For each attribute above, we will frequently use its first letter to refer to
it, to save space in the slides, unless it creates an ambiguity
Some ages do not make sense, but this is fine for our example
74. 74
Relational Implementation
Two options for selecting the primary key of Marriage
The design is not necessarily good, but nice and simple
for learning relational algebra
Because we want to focus on relational algebra, which
does not understand keys, we will not specify keys in
this unit
Marriage
PK,FK2 W
FK1 H
Person
PK N
S
A
Birth
PK,FK1 P
PK,FK2 C
Marriage
PK,FK1 H
FK2 W
Person
PK N
S
A
Birth
PK,FK1 P
PK,FK2 C
75. 75
Microsoft Access Database
Microsoft Access Database with this example
has been posted
» The example suggests that you download and install
Microsoft Access 2007
» The examples are in the Access 2000 format so that
if you have an older version, you can work with it
Access is a very good tool for quickly learning
basic constructs of SQL DML, although it is not
suitable for anything other than personal
databases
76. 76
Microsoft Access Database
The database and our queries (other than the one with
MINUS at the end) are the appropriate “extras” directory
on the class web in “slides”
» MINUS is frequently specified in commercial databases in a
roundabout way
» We will cover how it is done when we discuss commercial
databases
Our sample Access database: People.mdb
The queries in Microsoft Access are copied and pasted
in these notes, after reformatting them
Included copied and pasted screen shots of the results
of the queries so that you can correlate the queries with
the names of the resulting tables
78. 78
Our Database
Person N S A
Albert M 20
Dennis M 40
Evelyn F 20
John M 60
Mary F 40
Robert M 60
Susan F 40
Birth P C
Dennis Albert
John Mary
Mary Albert
Robert Evelyn
Susan Evelyn
Susan Richard
Marriage H W A
Dennis Mary 20
Robert Susan 30
80. 80
A Query
Produce the relation Answer(A) consisting of all
ages of people
Note that all the information required can be
obtained from looking at a single relation,
Person
Answer:=
SELECT A
FROM Person;
A
20
40
20
60
40
60
40
81. 81
The Query In Microsoft Access
The actual query was copied and pasted
from Microsoft Access and reformatted for
readability
The result is below
82. 82
A Query
Produce the relation Answer(N) consisting of all
women who are less or equal than 32 years old.
Note that all the information required can be
obtained from looking at a single relation,
Person
Answer:=
SELECT N
FROM Person
WHERE A <= 32 AND S =‘F’;
N
Evelyn
83. 83
The Query In Microsoft Access
The actual query was copied and pasted
from Microsoft Access and reformatted for
readability
The result is below
84. 84
A Query
Produce a relation Answer(P, Daughter) with
the obvious meaning
Here, even though the answer comes only from
the single relation Birth, we still have to check in
the relation Person what the S of the C is
To do that, we create the Cartesian product of
the two relations: Person and Birth. This gives
us “long tuples,” consisting of a tuple in Person
and a tuple in Birth
For our purpose, the two tuples matched if N in
Person is C in Birth and the S of the N is F
85. 85
A Query
Answer:=
SELECT P, C AS Daughter
FROM Person, Birth
WHERE C = N AND S = ‘F’;
P Daughter
John Mary
Robert Evelyn
Susan Evelyn
86. 86
Cartesian Product With Condition: Matching Tuples Indicated
Person N S A
Albert M 20
Dennis M 40
Evelyn F 20
John M 60
Mary F 40
Robert M 60
Susan F 40
Birth P C
Dennis Albert
John Mary
Mary Albert
Robert Evelyn
Susan Evelyn
Susan Richard
87. 87
The Query In Microsoft Access
The actual query was copied and pasted
from Microsoft Access and reformatted for
readability
The result is below
88. 88
A Query
Produce a relation Answer(Father, Daughter)
with the obvious meaning.
Here we have to simultaneously look at two
copies of the relation Person, as we have to
determine both the S of the Parent and the S of
the C
We need to have two distinct copies of
Person in our SQL query
But, they have to have different names so we
can specify to which we refer
Again, we use AS as a renaming operator,
these time for relations
89. 89
A Query
Answer :=
SELECT P AS Father, C AS Daughter
FROM Person, Birth, Person AS Person1
WHERE P = Person.N AND C = Person1.N
AND Person.S = ‘M’ AND Person1.S = ‘F’;
Father Daughter
John Mary
Robert Evelyn
90. 90
Cartesian Product With Condition: Matching Tuples Indicated
Person N S A
Albert M 20
Dennis M 40
Evelyn F 20
John M 60
Mary F 40
Robert M 60
Susan F 40
Birth P C
Dennis Albert
John Mary
Mary Albert
Robert Evelyn
Susan Evelyn
Susan Richard
Person N S A
Albert M 20
Dennis M 40
Evelyn F 20
John M 60
Mary F 40
Robert M 60
Susan F 40
91. 91
The Query In Microsoft Access
The actual query was copied and pasted
from Microsoft Access and reformatted for
readability
The result is below
92. 92
A Query
Produce a relation: Answer(Father_in_law,
Son_in_law).
A classroom exercise, but you can see the
solution in the posted database.
Hint: you need to compute the Cartesian
product of several relations if you start from
scratch, or of two relations if you use the
previously computed (Father, Daughter) relation
F_I_L S_I_L
John Dennis
93. 93
The Query In Microsoft Access
The actual query was copied and pasted
from Microsoft Access and reformatted for
readability
The result is below
94. 94
A Query
Produce a relation:
Answer(Grandparent,Grandchild)
A classroom exercise, but you can see
the solution in the posted database
G_P G_C
John Albert
95. 95
Cartesian Product With Condition: Matching Tuples Indicated
Birth P C
Dennis Albert
John Mary
Mary Albert
Robert Evelyn
Susan Evelyn
Susan Richard
Birth P C
Dennis Albert
John Mary
Mary Albert
Robert Evelyn
Susan Evelyn
Susan Richard
96. 96
The Query In Microsoft Access
The actual query was copied and pasted
from Microsoft Access and reformatted for
readability
The result is below
97. 97
Further Distance
How to compute (Great-grandparent,Great-grandchild)?
Easy, just take the Cartesian product of the
(Grandparent, Grandchild) table with (Parent,Child)
table and specify equality on the “intermediate” person
How to compute (Great-great-grandparent,Great-great-
grandchild)?
Easy, just take the Cartesian product of the
(Grandparent, Grandchild) table with itself and specify
equality on the “intermediate” person
Similarly, can compute (Greatx-grandparent,Greatx-
grandchild), for any x
Ultimately, may want (Ancestor,Descendant)
98. 98
Relational Algebra Is Not Universal:Cannot Compute (Ancestor,Descendant)
Standard programming languages are
universal
This roughly means that they are as powerful as
Turing machines, if unbounded amount of
storage is permitted (you will never run out of
memory)
This roughly means that they can compute
anything that can be computed by any
computational machine we can (at least
currently) imagine
Relational algebra is weaker than a standard
programming language
It is impossible in relational algebra (or standard
SQL) to compute the relation Answer(Ancestor,
Descendant)
99. 99
Relational Algebra Is Not Universal: Cannot Compute (Ancestor,Descendant)
It is impossible in relational algebra (or standard SQL)
to compute the relation Answer(Ancestor, Descendant)
Why?
The proof is a reasonably simple, but uses cumbersome
induction.
The general idea is:
» Any relational algebra query is limited in how many relations or
copies of relations it can refer to
» Computing arbitrary (ancestor, descendant) pairs cannot be
done, if the query is limited in advance as to the number of
relations and copies of relations (including intermediate results)
it can specify
This is not a contrived example because it shows that
we cannot compute the transitive closure of a directed
graph: the set of all paths in the graph
100. 100
A Sample Query
Produce a relation Answer(A) consisting
of all ages of visitors that are not ages of
marriages
SELECT
A FROM Person
MINUS
SELECT
A FROM MARRIAGE;
101. 101
The Query In Microsoft Access
We do not show this here, as it is done in
a roundabout way and we will do it later
102. 102
It Does Not Matter If We Remove Duplicates
Removing duplicates
- =
Not removing duplicates
- =
A
20
40
20
60
40
60
40
A
20
30
A
40
60
40
60
40
A
20
40
60
A
20
30
A
40
60
103. 103
It Does Not Matter If We Remove Duplicates
The resulting set contains precisely ages: 40, 60
So we do not have to be concerned with whether the
implementation removes duplicates from the result or not
In both cases we can answer correctly
» Is 50 a number that is an age of a marriage but not of a person
» Is 40 a number that is an age of a marriage but not of a person
Just like we do not have to be concerned with whether it
sorts (orders) the result
This is the consequence of us not insisting that an
element in a set appears only once, as we discussed
earlier
Note, if we had said that an element in a set appears
once, we would have to spend effort removing
duplicates!
104. 104
Now To “Pure” Relational Algebra
This was described in several slides
But it is really the same as before, just the
notation is more mathematical
Looks like mathematical expressions, not
snippets of programs
It is useful to know this because many
articles use this instead of SQL
This notation came first, before SQL was
invented, and when relational databases
were just a theoretical construct
105. 105
p: Projection: Choice Of Columns
SQL statement Relational
Algebra
SELECT B, A, D pB,A,D (R)
FROM R
We could have removed the duplicate row,
but did not have to
R A B C D
1 10 100 1000
1 20 100 1000
1 20 200 1000
B A D
10 1 1000
20 1 1000
20 1 1000
106. 106
s: Selection: Choice Of Rows
SQL statement: Relational Algebra
SELECT * sA C D=4 (R) Note: no need for p
FROM R
WHERE A <= C AND D = 4;
R A B C D
5 5 7 4
5 6 5 7
4 5 4 4
5 5 5 5
4 6 5 3
4 4 3 4
4 4 4 5
4 6 4 6
A B C D
5 5 7 4
4 5 4 4
107. 107
Selection
In general, the condition (predicate) can be
specified by a Boolean formula with
and on atomic conditions, where a
condition is:
» a comparison between two column names,
» a comparison between a column name and a
constant
» Technically, a constant should be put in quotes
» Even a number, such as 4, perhaps should be put in
quotes, as ‘4’ so that it is distinguished from a
column name, but as we will never use numbers for
column names, this not necessary
108. 108
: Cartesian Product
SQL statement Relational Algebra
SELECT A, R.B, C, S.B, D R S
FROM R, S
A R.B C S.B D
1 10 40 10 10
1 10 50 20 10
2 10 40 10 10
2 10 50 20 10
2 20 40 10 10
2 20 50 20 10
R A B
1 10
2 10
2 20
S C B D
40 10 10
50 20 10
109. 109
A Typical Use Of Cartesian Product
SQL statement: Relational Algebra
SELECT ID#, R.Room#, Size p ID#, R.Room#, Size (sR.B = S.B (R S))
FROM R, S
WHERE R.Room# = S.Room#
R Size Room#
140 1010
150 1020
140 1030
S ID# Room# YOB
40 1010 1982
50 1020 1985
ID# R.Room# Size
40 1010 140
50 1020 150
110. 110
: Union
SQL statement Relational Algebra
(SELECT * R S
FROM R)
UNION
(SELECT *
FROM S)
Note: We happened to choose to remove duplicate
rows
Union compatibility required
R A B
1 10
2 20
S A B
1 10
3 20
A B
1 10
2 20
3 20
111. 111
: Difference
SQL statement Relational
Algebra
(SELECT * R S
FROM R)
MINUS
(SELECT *
FROM S)
Union compatibility required
R A B
1 10
2 20
S A B
1 10
3 20
A B
2 20
112. 112
: Intersection
SQL statement Relational
Algebra
(SELECT * R S
FROM R)
INTERSECT
(SELECT *
FROM S)
Union compatibility required
R A B
1 10
2 20
S A B
1 10
3 20
A B
1 10
113. 113
Summary (1/2)
A relation is a set of rows in a table with labeled columns
Relational algebra as the basis for SQL
Basic operations:
» Union (requires union compatibility)
» Difference (requires union compatibility)
» Intersection (requires union compatibility); technically not a basic
operation
» Selection of rows
» Selection of columns
» Cartesian product
These operations define an algebra: given an expression
on relations, the result is a relation (this is a “closed”
system)
Combining this operations allows production of
sophisticated queries
114. 114
Summary (2/2)
Relational algebra is not universal: We can write
programs producing queries that cannot be
specified using relational algebra
We focused on relational algebra specified
using SQL syntax, as this is more important in
practice
The other, “more mathematical” notation came
first and is used in research and other venues,
but not commercially
115. 115
Agenda
1 Session Overview
4 Summary and Conclusion
2 Relational Algebra and Relational Calculus
3 Relational Algebra Using SQL Syntax
116. 116
Summary
Relational algebra and relational calculus are formal
languages for the relational model of data
A relation is a set of rows in a table with labeled columns
Relational algebra and associated operations are the
basis for SQL
These relational operations define an algebra
Relational algebra is not universal as it is possible to write
programs producing queries that cannot be specified
using relational algebra
Relational algebra can be specified using SQL syntax
The other, “more mathematical” notation came first and is
used in research and other venues, but not commercially
117. 117
Assignments & Readings
Readings
» Slides and Handouts posted on the course web site
» Textbook: Chapters 6
Assignment #3
» Textbook exercises: Textbook exercises: 9.3, 9.5, 10.23, 6.16, 6.24, 6.32
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