Archive for relational algebra

Database Terminologies

Database


A database is a logically coherent collection of data with some inherent meaning, representing some aspect of real world and which is designed, built and populated with data for a specific purpose.
   

DBMS


It is a collection of programs that enables user to create and maintain a database. In other words it is general-purpose software that provides the users with the processes of defining, constructing and manipulating the database for various applications.
   

Database system

The database and DBMS software together is called as Database system.

   

Advantages of DBMS
  • Redundancy is controlled.
  • Unauthorised access is restricted.
  • Providing multiple user interfaces.
  • Enforcing integrity constraints.
  • Providing backup and recovery.

 

   

Disadvantage in File Processing System
  • Data redundancy & inconsistency.
  • Difficult in accessing data.
  • Data isolation.
  • Data integrity.
  • Concurrent access is not possible.
  • Security Problems.

   

three levels of abstraction:
  • Physical level: The lowest level of abstraction describes how data are stored.
  • Logical level: The next higher level of abstraction, describes what data are stored in database and what relationship among those data.
  • View level: The highest level of abstraction describes only part of entire database.

   

integrity rules

There are two Integrity rules.

  • Entity Integrity: States that “Primary key cannot have NULL value”
  • Referential Integrity: States that “Foreign Key can be either a NULL value or should be Primary Key value of other relation.

   

extension and intension


Extension -

It is the number of tuples present in a table at any instance. This is time dependent.

Intension -

It is a constant value that gives the name, structure of table and the constraints laid on it.

 

   

System R and its two major subsystems

System R was designed and developed over a period of – at IBM San Jose Research Center. It is a prototype and its purpose was to demonstrate that it is possible to build a Relational System that can be used in a real life environment to solve real life problems, with performance at least comparable to that of existing system.
Its two subsystems are

  • Research Storage
  • System Relational Data System.

   

data structure of System R


Unlike Relational systems in System R

  • Domains are not supported
  • Enforcement of candidate key uniqueness is optional
  • Enforcement of entity integrity is optional
  • Referential integrity is not enforced

   

Data Independence

Data independence means that “the application is independent of the storage structure and access strategy of data”. In other words, The ability to modify the schema definition in one level should not affect the schema definition in the next higher level.
Two types of Data Independence:

  • Physical Data Independence: Modification in physical level should not affect the logical level.
  • Logical Data Independence: Modification in logical level should affect the view level.

NOTE: Logical Data Independence is more difficult to achieve

   

view and data independence

A view may be thought of as a virtual table, that is, a table that does not really exist in its own right but is instead derived from one or more underlying base table. In other words, there is no stored file that direct represents the view instead a definition of view is stored in data dictionary.
Growth and restructuring of base tables is not reflected in views. Thus the view can insulate users from the effects of restructuring and growth in the database. Hence accounts for logical data independence.

   

Data Model

A collection of conceptual tools for describing data, data relationships data semantics and constraints.

   

E-R model

This data model is based on real world that consists of basic objects called entities and of relationship among these objects. Entities are described in a database by a set of attributes.

   

Object Oriented model

This model is based on collection of objects. An object contains values stored in instance variables with in the object. An object also contains bodies of code that operate on the object. These bodies of code are called methods. Objects that contain same types of values and the same methods are grouped together into classes.

   

Entity

It is a ‘thing’ in the real world with an independent existence.

   

Entity type

It is a collection (set) of entities that have same attributes.

   

Entity set

It is a collection of all entities of particular entity type in the database.

Extension of entity type

The collections of entities of a particular entity type are grouped together into an entity set.

   

Weak Entity set

An entity set may not have sufficient attributes to form a primary key, and its primary key compromises of its partial key and primary key of its parent entity, then it is said to be Weak Entity set.

an attribute

It is a particular property, which describes the entity.

   

Relation Schema and a Relation

A relation Schema denoted by R(A, A, …, An) is made up of the relation name R and the list of attributes Ai that it contains. A relation is defined as a set of tuples. Let r be the relation which contains set tuples (t, t, t, …, tn). Each tuple is an ordered list of n-values t=(v,v, …, vn).

degree of a Relation

It is the number of attribute of its relation schema.

Relationship

It is an association among two or more entities.

Relationship set

The collection (or set) of similar relationships.

Relationship type

Relationship type defines a set of associations or a relationship set among a given set of entity types.

degree of Relationship type

It is the number of entity type participating.

   

DDL (Data Definition Language)

A data base schema is specifies by a set of definitions expressed by a special language called DDL.

   

VDL (View Definition Language)

It specifies user views and their mappings to the conceptual schema.

   

SDL (Storage Definition Language)

This language is to specify the internal schema. This language may specify the mapping between two schemas.

   

Data Storage - Definition Language

The storage structures and access methods used by database system are specified by a set of definition in a special type of DDL called data storage-definition language.

   

DML (Data Manipulation Language)

This language that enable user to access or manipulate data as organised by appropriate data model.

  • Procedural DML or Low level: DML requires a user to specify what data are needed and how to get those data.
  • Non-Procedural DML or High level: DML requires a user to specify what data are needed without specifying how to get those data.

   

DML Compiler

It translates DML statements in a query language into low-level instruction that the query evaluation engine can understand.

   

Query evaluation engine

It executes low-level instruction generated by compiler.

   

DDL Interpreter

It interprets DDL statements and record them in tables containing metadata.

   

Record-at-a-time

The Low level or Procedural DML can specify and retrieve each record from a set of records. This retrieve of a record is said to be Record-at-a-time.

   

Set-at-a-time or Set-oriented

The High level or Non-procedural DML can specify and retrieve many records in a single DML statement. This retrieve of a record is said to be Set-at-a-time or Set-oriented.

   

Relational Algebra

It is procedural query language. It consists of a set of operations that take one or two relations as input and produce a new relation.

   

Relational Calculus

It is an applied predicate calculus specifically tailored for relational databases proposed by E.F. Codd. E.g. of languages based on it are DSL ALPHA, QUEL.

Tuple-oriented relational calculus differ from domain-oriented relational calculus

The tuple-oriented calculus uses a tuple variables i.e., variable whose only permitted values are tuples of that relation. E.g. QUEL
The domain-oriented calculus has domain variables i.e., variables that range over the underlying domains instead of over relation. E.g. ILL, DEDUCE.

   

normalization and normal forms

It is a process of analysing the given relation schemas based on their Functional Dependencies (FDs) and primary key to achieve the properties

  • Minimizing redundancy
  • Minimizing insertion, deletion and update anomalies.

Functional Dependency

A Functional dependency is denoted by X Y between two sets of attributes X and Y that are subsets of R specifies a constraint on the possible tuple that can form a relation state r of R. The constraint is for any two tuples t and t in r if t[X] = t[X] then they have t[Y] = t[Y]. This means the value of X component of a tuple uniquely determines the value of component Y.

functional dependency F said to be minimal when

  • Every dependency in F has a single attribute for its right hand side.
  • We cannot replace any dependency X A in F with a dependency Y A where Y is a proper subset of X and still have a set of dependency that is equivalent to F.
  • We cannot remove any dependency from F and still have set of dependency that is equivalent to F.

Multivalued dependency

Multivalued dependency denoted by X Y specified on relation schema R, where X and Y are both subsets of R, specifies the following constraint on any relation r of R: if two tuples t and t exist in r such that t[X] = t[X] then t and t should also exist in r with the following properties

  • t[x] = t[X] = t[X] = t[X]
  • t[Y] = t[Y] and t[Y] = t[Y]
  • t[Z] = t[Z] and t[Z] = t[Z]

where [Z = (R-(X U Y)) ]

Lossless join property

It guarantees that the spurious tuple generation does not occur with respect to relation schemas after decomposition.

NF (Normal Form)

The domain of attribute must include only atomic (simple, indivisible) values.

Fully Functional dependency

It is based on concept of full functional dependency. A functional dependency X Y is full functional dependency if removal of any attribute A from X means that the dependency does not hold any more.

First NF

A relation schema R is in NF if it is in NF and every non-prime attribute A in R is fully functionally dependent on primary key.

Second NF

A relation schema R is in second NF if it is in first NF and for every FD X A either of the following is true

  • X is a Super-key of R.
  • A is a prime attribute of R.

In other words, if every non prime attribute is non-transitively dependent on primary key.

BCNF (Boyce-Codd Normal Form)

A relation schema R is in BCNF if it is in NF and satisfies an additional constraint that for every FD X A, X must be a candidate key.

Fourth NF

A relation schema R is said to be in NF if for every Multivalued dependency X Y that holds over R, one of following is true

  • X is subset or equal to (or) XY = R.
  • X is a super key.

Fifth NF

A Relation schema R is said to be NF if for every join dependency {R, R, …, Rn} that holds R, one the following is true

  • Ri = R for some i.
  • The join dependency is implied by the set of FD, over R in which the left side is key of R.

Domain-Key Normal Form

A relation is said to be in DKNF if all constraints and dependencies that should hold on the the constraint can be enforced by simply enforcing the domain constraint and key constraint on the relation.

   

Keys – Partial, alternate, compound and artificial

It is a set of attributes that can uniquely identify weak entities and that are related to same owner entity. It is sometime called as Discriminator.

Alternate Key:

All Candidate Keys excluding the Primary Key are known as Alternate Keys.

Artificial Key:

If no obvious key, either stand alone or compound is available, then the last resort is to simply create a key, by assigning a unique number to each record or occurrence. Then this is known as developing an artificial key.

Compound Key:

If no single data element uniquely identifies occurrences within a construct, then combining multiple elements to create a unique identifier for the construct is known as creating a compound key.

Natural Key:

When one of the data elements stored within a construct is utilized as the primary key, then it is called the natural key.

 

   

Indexing


Indexing is a technique for determining how quickly specific data can be found.
Types:

  • Binary search style indexing
  • B-Tree indexing
  • Inverted list indexing
  • Memory resident table
  • Table indexing

system catalog or catalog relation How is better known as

A RDBMS maintains a description of all the data that it contains, information about every relation and index that it contains. This information is stored in a collection of relations maintained by the system called metadata. It is also called data dictionary.

   

query optimization

The phase that identifies an efficient execution plan for evaluating a query that has the least estimated cost is referred to as query optimization.

   

Join Dependency

A Join dependency is generalization of Multivalued dependency.A JD {R, R, …, Rn} is said to hold over a relation R if R, R, R, …, Rn is a lossless-join decomposition of R . There is no set of sound and complete inference rules for JD.

Inclusion Dependency:

An Inclusion Dependency is a statement of the form that some columns of a relation are contained in other columns. A foreign key constraint is an example of inclusion dependency.

 

   

durability in DBMS

Once the DBMS informs the user that a transaction has successfully completed, its effects should persist even if the system crashes before all its changes are reflected on disk. This property is called durability.

   

Atomicity

Either all actions are carried out or none are. Users should not have to worry about the effect of incomplete transactions. DBMS ensures this by undoing the actions of incomplete transactions.

   

Aggregation:

A concept which is used to model a relationship between a collection of entities and relationships. It is used when we need to express a relationship among relationships.

 

   

Phantom Deadlock

In distributed deadlock detection, the delay in propagating local information might cause the deadlock detection algorithms to identify deadlocks that do not really exist. Such situations are called phantom deadlocks and they lead to unnecessary aborts.

   

checkpoint


A Checkpoint is like a snapshot of the DBMS state. By taking checkpoints, the DBMS can reduce the amount of work to be done during restart in the event of subsequent crashes.

   

Different phases of transaction are
  • Analysis phase
  • Redo Phase
  • Undo phase

   

flat file database

It is a database in which there are no programs or user access languages. It has no cross-file capabilities but is user-friendly and provides user-interface management.

   

transparent DBMS

It is one, which keeps its Physical Structure hidden from user.

Network, Hierarchical schemas and their properties

Network schema uses a graph data structure to organize records example for such a database management system is CTCG while a hierarchical schema uses a tree data structure example for such a system is IMS.

   

query

A query with respect to DBMS relates to user commands that are used to interact with a data base. The query language can be classified into data definition language and data manipulation language.

   

Correlated subquery

Subqueries, or nested queries, are used to bring back a set of rows to be used by the parent query. Depending on how the subquery is written, it can be executed once for the parent query or it can be executed once for each row returned by the parent query. If the subquery is executed for each row of the parent, this is called a correlated subquery.
A correlated subquery can be easily identified if it contains any references to the parent subquery columns in its WHERE clause. Columns from the subquery cannot be referenced anywhere else in the parent query. The following example demonstrates a non-correlated subquery.
E.g. Select * From CUST Where ‘// IN (Select ODATE From ORDER Where CUST.CNUM = ORDER.CNUM)

Addition, deletion and modification are the primitive operations common to all record management systems

‘Edit’ Buffer all the commands that are typed in are stored

unary operations in Relational Algebra

   

PROJECTION and SELECTION.

Are the resulting relations of PRODUCT and JOIN operation the same

No.
PRODUCT: Concatenation of every row in one relation with every row in another.
JOIN: Concatenation of rows from one relation and related rows from another.

   

RDBMS KERNEL

Two important pieces of RDBMS architecture are the kernel, which is the software, and the data dictionary, which consists of the system-level data structures used by the kernel to manage the database
You might think of an RDBMS as an operating system (or set of subsystems), designed specifically for controlling data access; its primary functions are storing, retrieving, and securing data. An RDBMS maintains its own list of authorized users and their associated privileges; manages memory caches and paging; controls locking for concurrent resource usage; dispatches and schedules user requests; and manages space usage within its table-space structures.

   

the sub-systems of a RDBMS

I/O, Security, Language Processing, Process Control, Storage Management, Logging and Recovery, Distribution Control, Transaction Control, Memory Management, Lock Management

   

data dictionary


Data dictionary is a set of tables and database objects that is stored in a special area of the database and maintained exclusively by the kernel.

the job of the information stored in data-dictionary

The information in the data dictionary validates the existence of the objects, provides access to them, and maps the actual physical storage location.

   

SQL

SQL and the differences between SQL and other conventional programming Languages

SQL is a nonprocedural language that is designed specifically for data access operations on normalized relational database structures. The primary difference between SQL and other conventional programming languages is that SQL statements specify what data operations should be performed rather than how to perform them.

   

set of files on disk that compose a database in Oracle

There are three major sets of files on disk that compose a database. All the files are binary. These are

  • Database files
  • Control files
  • Redo logs

The most important of these are the database files where the actual data resides. The control files and the redo logs support the functioning of the architecture itself.
All three sets of files must be present, open, and available to Oracle for any data on the database to be useable. Without these files, you cannot access the database, and the database administrator might have to recover some or all of the database using a backup, if there is one.

   

an Oracle Instance

The Oracle system processes, also known as Oracle background processes, provide functions for the user processes—functions that would otherwise be done by the user processes themselves
Oracle database-wide system memory is known as the SGA, the system global area or shared global area. The data and control structures in the SGA are shareable, and all the Oracle background processes and user processes can use them.
The combination of the SGA and the Oracle background processes is known as an Oracle instance

the four Oracle system processes that must always be up and running for the database to be useable

The four Oracle system processes that must always be up and running for the database to be useable include DBWR (Database Writer), LGWR (Log Writer), SMON (System Monitor), and PMON (Process Monitor).

database files, control files and log files.

  • Database Files
    The database files hold the actual data and are typically the largest in size. Depending on their sizes, the tables (and other objects) for all the user accounts can go in one database file—but that’s not an ideal situation because it does not make the database structure very flexible for controlling access to storage for different users, putting the database on different disk drives, or backing up and restoring just part of the database.
    You must have at least one database file but usually, more than one files are used. In terms of accessing and using the data in the tables and other objects, the number (or location) of the files is immaterial.
    The database files are fixed in size and never grow bigger than the size at which they were created
  • Control Files
    The control files and redo logs support the rest of the architecture. Any database must have at least one control file, although you typically have more than one to guard against loss. The control file records the name of the database, the date and time it was created, the location of the database and redo logs, and the synchronization information to ensure that all three sets of files are always in step. Every time you add a new database or redo log file to the database, the information is recorded in the control files.
  • Redo Logs
    Any database must have at least two redo logs. These are the journals for the database; the redo logs record all changes to the user objects or system objects. If any type of failure occurs, the changes recorded in the redo logs can be used to bring the database to a consistent state without losing any committed transactions. In the case of non-data loss failure, Oracle can apply the information in the redo logs automatically without intervention from the DBA.
    The redo log files are fixed in size and never grow dynamically from the size at which they were created.

   

ROWID

The ROWID is a unique database-wide physical address for every row on every table. Once assigned (when the row is first inserted into the database), it never changes until the row is deleted or the table is dropped.
The ROWID consists of the following three components, the combination of which uniquely identifies the physical storage location of the row.

  • Oracle database file number, which contains the block with the rows
  • Oracle block address, which contains the row
  • The row within the block (because each block can hold many rows)

The ROWID is used internally in indexes as a quick means of retrieving rows with a particular key value. Application developers also use it in SQL statements as a quick way to access a row once they know the ROWID

   

.iOracle Block

Oracle “formats” the database files into a number of Oracle blocks when they are first created—making it easier for the RDBMS software to manage the files and easier to read data into the memory areas.
The block size should be a multiple of the operating system block size. Regardless of the block size, the entire block is not available for holding data; Oracle takes up some space to manage the contents of the block. This block header has a minimum size, but it can grow.
These Oracle blocks are the smallest unit of storage. Increasing the Oracle block size can improve performance, but it should be done only when the database is first created.
Each Oracle block is numbered sequentially for each database file starting at . Two blocks can have the same block address if they are in different database files.

   

database Trigger

A database trigger is a PL/SQL block that can defined to automatically execute for insert, update, and delete statements against a table. The trigger can e defined to execute once for the entire statement or once for every row that is inserted, updated, or deleted. For any one table, there are twelve events for which you can define database triggers. A database trigger can call database procedures that are also written in PL/SQL.

   

Backup and recovery. Of Oracle

Along with the RDBMS software, Oracle provides two utilities that you can use to back up and restore the database. These utilities are Export and Import.
The Export utility dumps the definitions and data for the specified part of the database to an operating system binary file. The Import utility reads the file produced by an export, recreates the definitions of objects, and inserts the data
If Export and Import are used as a means of backing up and recovering the database, all the changes made to the database cannot be recovered since the export was performed. The best you can do is recover the database to the time when the export was last performed.

.

   

stored-procedures and advantages of using them.

Stored procedures are database objects that perform a user defined operation. A stored procedure can have a set of compound SQL statements. A stored procedure executes the SQL commands and returns the result to the client. Stored procedures are used to reduce network traffic.

   

exceptions handling in PL/SQL

PL/SQL exception handling is a mechanism for dealing with run-time errors encountered during procedure execution. Use of this mechanism enables execution to continue if the error is not severe enough to cause procedure termination.
The exception handler must be defined within a subprogram specification. Errors cause the program to raise an exception with a transfer of control to the exception-handler block. After the exception handler executes, control returns to the block in which the handler was defined. If there are no more executable statements in the block, control returns to the caller.
User-Defined Exceptions
PL/SQL enables the user to define exception handlers in the declarations area of subprogram specifications. User accomplishes this by naming an exception

In addition to this list of exceptions, there is a catch-all exception named OTHERS that traps all errors for which specific error handling has not been established.

   

“PL/SQL

The concept of overloading in PL/SQL relates to the idea that you can define procedures and functions with the same name. PL/SQL does not look only at the referenced name, however, to resolve a procedure or function call. The count and data types of formal parameters are also considered.
PL/SQL also attempts to resolve any procedure or function calls in locally defined packages before looking at globally defined packages or internal functions. To further ensure calling the proper procedure, you can use the dot notation. Prefacing a procedure or function name with the package name fully qualifies any procedure or function reference.

   

It is a program module that provides the interface between the low-level data stored in database, application programs and queries submitted to the system.

   

Buffer Manager

It is a program module, which is responsible for fetching data from disk storage into main memory and deciding what data to be cache in memory.

   

Transaction Manager

It is a program module, which ensures that database, remains in a consistent state despite system failures and concurrent transaction execution proceeds without conflicting.

   

File Manager

It is a program module, which manages the allocation of space on disk storage and data structure used to represent information stored on a disk.

   

Authorization and Integrity manager

It is the program module, which tests for the satisfaction of integrity constraint and checks the authority of user to access data.

   

stand-alone procedures

Procedures that are not part of a package are known as stand-alone because they independently defined. A good example of a stand-alone procedure is one written in a SQL*Forms application. These types of procedures are not available for reference from other Oracle tools. Another limitation of stand-alone procedures is that they are compiled at run time, which slows execution.

   

different types of cursors.

PL/SQL uses cursors for all database information accesses statements. The language supports the use two types of cursors

  • Implicit
  • Explicit

   

cold backup and hot backup (in case of Oracle)
  • Cold Backup:
    It is copying the three sets of files (database files, redo logs, and control file) when the instance is shut down. This is a straight file copy, usually from the disk directly to tape. You must shut down the instance to guarantee a consistent copy.
    If a cold backup is performed, the only option available in the event of data file loss is restoring all the files from the latest backup. All work performed on the database since the last backup is lost.
  • Hot Backup:
    Some sites (such as worldwide airline reservations systems) cannot shut down the database while making a backup copy of the files. The cold backup is not an available option.
    So different means of backing up database must be used — the hot backup. Issue a SQL command to indicate to Oracle, on a tablespace-by-tablespace basis, that the files of the tablespace are to backed up. The users can continue to make full use of the files, including making changes to the data. Once the user has indicated that he/she wants to back up the tablespace files, he/she can use the operating system to copy those files to the desired backup destination.
    The database must be running in ARCHIVELOG mode for the hot backup option.
    If a data loss failure does occur, the lost database files can be restored using the hot backup and the online and offline redo logs created since the backup was done. The database is restored to the most consistent state without any loss of committed transactions.

   

Proactive, Retroactive and Simultaneous Update
  • Proactive Update:
    The updates that are applied to database before it becomes effective in real world .
  • Retroactive Update:
    The updates that are applied to database after it becomes effective in real world .
  • Simulatneous Update:
    The updates that are applied to database at the same time when it becomes effective in real world .

   
 

Relational Algebra

query language is a language in which user requests information from the database. it can be categorized as either procedural or nonprocedural.In a procedural language the user instructs the system to do a sequence of operations on database to compute the desired result. In nonprocedural language the user describes the desired information without giving a specific procedure for obtaining that information.

The relational algebra is a procedural query language. It consists of a set of operations that take one or two relations as input and produces a new relation as output.

Fundamental Operations

 

  • SELECT
  • PROJECT
  • UNION
  • SET DIFFERENCE
  • CARTESIAN PRODUCT
  • RENAME

Select and project operations are unary operation as they operate on a single relation.Union, set difference, Cartesian product and rename operations are binary operations as they operate on pairs of relations.

Other Operations

 

  • SET INTERSECTION
  • NATURAL JOIN
  • DIVISION
  • ASSIGNMENT

The select operation: -

to identify a set of tuples which is a part of a relation and to extract only these tuples out. The select operation selects tuples that satisfy a given predicate or condition.

  • It is a unary operation defined on a single relation.
  • It is denoted as σ.

Consider the following table “Book” :-

Code:

+--------+--------+------------------+
| Acc-no | Yr-pub | title            |
+--------+--------+------------------+
| 734216 | 1982   | Algorithm design |
| 237235 | 1995   | Database systems |
| 631523 | 1992   | Compiler design  |
| 543211 | 1991   | Programming      |
| 376112 | 1992   | Machine design   |
+--------+--------+------------------+

Example1:- Select from the relation “Book” all the books whose year of publication is 1992.

Code:

σ Yr-pub=1992(Book)

Example2:- Select from the relation “Book” all the books whose Acc-no is greater than equal to 56782.

Code:

σ  Acc-no>=56782(Book)

The project operation: -

returns its argument relation with certain attributes left out.

  • It is a unary operation defined on a single relation
  • It is denoted as Π.

Example:- List all the Title and Acc-no of the “Book” relation.

Code:

   Π Acc-no, Title (Book)

The union operation: -

is used when we need some attributes that appear in either or both of the two relations.

  • It is denoted as U.

Example:
Borrower (customer-name, loan-number)
Depositor (customer-name, account-number)
Customer (customer-name, street-number, customer-city)

List all the customers who have either an account or a loan or both

Code:

  Π customer-name (Borrower) U Π customer-name (Depositor)

For a union operation r U s to be valid, two conditions must hold:

  • The relation r and s must be of the same arity, i.e. they must have the same number of attributes.
  • The domains of the ith attribute of r and the ith attribute of s must be the same for all i.

The set difference operation: - 

finds tuples in one relation but not in other.

  • It is denoted as 

Example:
Find the names of all customers who have an account but not a loan.

Code:

  Π customer-name (Depositor) - Π customer-name (Borrower)

The Cartesian product operation: - 

allows combining information from two relations.

  • It is denoted as r X s where r and s are relations.

Consider the following relation or table “r” :-

Code:

+--------+-------+
A     | B     |
+--------+-------+
|   a    |   1   |
|   b    |   2   |
|   a    |   2   |
+--------+-------+

Consider another relation or table “s” :-

Code:

+--------+-------+
B     | C     |
+--------+-------+
|   3    |   1a  |
|   2    |   2b  |
+--------+-------+

Therefore, rXs gives:-

Code:

+-----------+---------+---------+----------+
|  r.A      |r.B      s.B    | s.C      |
+-----------+---------+---------+----------+
|   a       |   1     |    3    |  1a      |
|   a       |   1     |    2    |  2b      |
|   b       |   2     |    3    |  1a      |
|   b       |   2     |    2    |  2b      |
|   a       |   2     |    3    |  1a      |
|   a       |   2     |    2    |  2b      |
+-----------+---------+---------+----------+

If relation r has n1 tuples and relation s has n2 tuples then r X s has n1*n2 tuples.

Example:
Borrower (customer-name, loan-number)
Loan (loan-number, branch-name, city, amount)

List the names of all customers who have a loan in “Perryridge” branch

Code:

  Π customer-name (σ Borrower.loan-number=Loan.loan-number (σ branch-name=”Perryridge” (Borrower X Loan)))

The rename operation: -

 used to rename.

  • It is denoted as ρ.

: relational algebra expression
ρ x (E): returns the result of expression E under the name x.
ρ x (A1, A2, A3… An) (E): returns the result of expression E under the name x with attributes renamed to A1, A2, A3… An.


The set intersection operation: - 

finds tuples in both the relations.

  • It is denoted as .

Example:
Borrower (customer-name, loan-number)
Depositor (customer-name, account-number)
Customer (customer-name, street-number, customer-city)

List all the customers who have both a loan and an account.

Code:

  Π customer-name (Borrower) ∩ Π customer-name (Depositor)


The natural join operation: - 

it is a binary operation and a combination of certain selections and a Cartesian product into one operation.

  • It is denoted as |X| .
  • It is associative.

It forms a Cartesian product of its two arguments.
Then performs a selection forcing equality on those attributes those appear in both the relations.
And finally removes duplicates attributes.

r(R): r is a relation with attributes R.
s(S): s is a relation with attributes S.

If R  S = Ф i.e. they have no attributes in common then r |X| s = r X s

Example:-
Table “r”:-

Code:

+--------+--------+-------+
| A      | B      | C     |
+--------+--------+-------+
| a      |  b     | c     |
| d      |  e     | f     |
| g      |  h     | i     |
+--------+--------+-------+

Table “s” :-

Code:

+--------+-------+
B     | D     |
+--------+-------+
|   b    |   g   |
|   p    |   r   |
|   e    |   t   |
+--------+-------+

Therefore, r |X| s :-

Code:

+-----------+---------+---------+----------+
| A         |  B      |    C    |       |
+-----------+---------+---------+----------+
|   a       |   b     |    c    |  g       |
|   d       |   e     |    f    |  t       |
+-----------+---------+---------+----------+

i.e;
if r (A, B, C), s (B, D) then

Code:

   r|X|s = Π r.A, r.B, r.C, s.D (σ r.B = s.B (r X s))

The division / quotient operation: - 

  • It is denoted as ÷.

Letr(R) and s(S) be relations

r ÷ s: - the result consists of the restrictions of tuples in r to the attribute names unique to R, i.e. in the Header of r but not in the Header of s, for which it holds that all their combinations with tuples in s are present in r.

Example:

Relation or table “r”:-

Code:

+--------+-------+
A     | B     |
+--------+-------+
|   a    |   1   |
|   b    |   2   |
|   a    |   2   |
|   p    |   3   |
|   p    |   4   |
+--------+-------+

Relation or table “s”:-

Code:

+------+
|  B   |
+------+
|  2   |
|  3   |
+------+

Therefore, r ÷ s

Code:

+------+
|  A   |
+------+
|  b   |
|  a   |
|  p   |
+------+

Extended Relational Algebra Operations

GENERALIZED PROJECTION: -

It extends the projection operation by allowing arithmetic functions to be used in projection list.

Π F1,F2 … Fn (E)

Where E: relational algebra expression
Fi: arithmetic expression

Example:

Table “Credit-info” :-

Code:

+----------------------+-----------+-----------------+
| Customer-name        | Limit     | Credit_Balance  |
+----------------------+-----------+-----------------+
| abc                  | 2000      | 500             |
| xyz                  | 500       | 250             |
| pqr                  | 700       | 100             |
| mno                  | 1500      | 1000            |
+----------------------+-----------+-----------------+

Find how much money a person can spend.

Code:

  Π Customer-name,(Limit-Credit_balance)(Credit-info) 

AGGREGATE FUNCTION:-

It takes a collection of values and returns a single value as a result.

Table “Record” :-

Code:

+----------------------+-----------+-----------------+
| Student              | Marks     | Address         |
+----------------------+-----------+-----------------+
| abc                  | 20        | Garia           |
| xyz                  | 50        | Behala          |
| pqr                  | 70        | Hindmotor       |
| mno                  | 15        | Garia           |
+----------------------+-----------+-----------------+

So the Aggregate Functions are:-

Code:

   G sum (Marks) (Record) = returns sum total of the Marks attribute of Record.

   G average (Marks) (Record) = returns average of the Marks attribute of Record.
   
   G min (Marks) (Record) = returns the minimum of the Marks attribute of Record.
  
   G max (Marks) (Record) = returns the maximum of the Marks attribute of Record.

   G count distinct (Address) (Record) = returns the number of distinct values of Address attribute of Record.

Let’s assume that we have a table named Account with three columns, namely Account_Number, Branch_Name and Balance.

We wish to find the maximum balance of each branch.

Code:


  Branch_Name G Max(Balance)(Account).

To find the highest balance of all accounts regardless of branch

Code:

G Max(Balance)(Account).

Limitations Of Relational Algebra

Although relational algebra seems powerful enough for most practical purposes, there are some simple and natural operators on relations which cannot be expressed by relational algebra. The transitive closure of a binary relation is one of them.