dr. mohamed hegazi1 the relational algebra and relational calculus

Dr. Mohamed Hegazi 1

The Relational Algebra and

Relational Calculus


• Unary Relational Operations

• Relational Algebra Operations from Set Theory

• Binary relational Operations

Outlines


The relational algebra is a set of operators that The relational algebra is a set of operators that take and return relationstake and return relations

• Unary operations take one relation,and return one relation:

– SELECT operation

– PROJECT operation

– Sequences of unary operations

– RENAME operation

Unary Relational Operations


ID ENMAE BDATE ADRESS SEX SALARY SUPID DNO1001 Ahmed 1984 Sohar M 800 1002 31002 Ali 1980 Sohar M 1100 1004 31003 Hind 1988 Muscat F 500 1002 11004 Omer 1978 Sohar M 1200 1008 11005 Salwa 1985 Sohar F 600 1002 31006 Issa 1981 Muscat M 1000 1004 11007 Noura 1986 Muscat F 500 1002 1

ID ENMAE BDATE ADRESS SEX SALARY SUPID DNO1001 Ahmed 1984 Sohar M 800 1002 31002 Ali 1980 Sohar M 1100 1004 31005 Salwa 1985 Sohar F 600 1002 3

Employee:

(DNO=3) (EMPLOYEE).

The The SELECTSELECT Operation Operation


• It selects a subset of tuples from a relation that satisfy a SELECT condition.

• The SELECT operation is denoted by: <Selection condition> (R)

• The degree of the relation resulting from a SELECT operation is the same as that of R

• For any selection-condition c, we have | <c> (R) | | R |

• The SELECT operation is commutative, that is, <c1> ( <c2> ( R )) = <c2> ( <c1> ( R ))

The SELECT Operation



ID ENMAE BDATE

ADRESS SEX SALARY SUPID DNO

1003 Hind 1988 Muscat F 500 1002 11005 Salwa 1985 Sohar F 600 1002 31007 Noura 1986 Muscat F 500 1002 1

Employee:

(SEX=F) (EMPLOYEE).



ID ENMAE BDATE ADRESS SEX SALARY SUPID DNO1003 Hind 1988 Muscat F 500 1002 11007 Noura 1986 Muscat F 500 1002 1

Employee:

(SEX=F AND DNO=1 ) (EMPLOYEE).


ID ENMAE BDATE ADRESS SEX SALARY SUPID DNO

1001 Ahmed 1984 Sohar M 800 1002 3

1002 Ali 1980 Sohar M 1100 1004 3

1003 Hind 1988 Muscat F 500 1002 1

1004 Omer 1978 Sohar M 1200 1008 1

1005 Salwa 1985 Sohar F 600 1002 3

1006 Issa 1981 Muscat M 1000 1004 1

1007 Noura 1986 Muscat F 500 1002 1

ENMAE SALARY DNO

Ahmed 800 3

Ali 1100 3

Hind 500 1

Omer 1200 1

Salwa 600 3

Issa 1000 1

Noura 500 1

Employee:

ENAME, SALARY,DNO (EMPLOYEE)

The PROJECT Operation


• The PROJECT operation , selects certain columns from a relation and discards the columns that are not in the PROJECT list

• The PROJECT operation is denoted by: <attribute list> ( R )

• Where attribute-list is a list of attributes from the relation R• The degree of the relation resulting from a PROJECT

operation is equal to the number of attributes in attribute-list.

• The PROJECT operation is not commutative

The PROJECT Operation



1001 Ahmed 1984 Sohar M 800 1002 3

1002 Ali 1980 Sohar M 1100 1004 3

1003 Hind 1988 Muscat F 500 1002 1

1004 Omer 1978 Sohar M 1200 1008 1

1005 Salwa 1985 Sohar F 600 1002 3

1006 Issa 1981 Muscat M 1000 1004 1

1007 Noura 1986 Muscat F 500 1002 1

ID ENMAE BDATE SALARY DNO

1001 Ahmed 1984 800 3

1002 Ali 1980 1100 3

1003 Hind 1988 500 1

1004 Omer 1978 1200 1

1005 Salwa 1985 600 3

1006 Issa 1981 1000 1

1007 Noura 1986 500 1

Employee:

ID,ENAME, BDATE,SALARY,DNO (EMPLOYEE)



1001 Ahmed 1984 Sohar M 800 1002 3

1002 Ali 1980 Sohar M 1100 1004 3

1003 Hind 1988 Muscat F 500 1002 1

1004 Omer 1978 Sohar M 1200 1008 1

1005 Salwa 1985 Sohar F 600 1002 3

1006 Issa 1981 Muscat M 1000 1004 1

1007 Noura 1986 Muscat F 500 1002 1

ENMAE SALARY

Ahmed 800

Ali 1100

Salwa 600

Employee:

ENAME, SALARY((DNO=3) (EMPLOYEE))

Find the name and salary of all employees who work on department number 3 (DNO=3)

The result :


RESULT =RESULT1 RESULT2.

Relational Algebra OperationsRelational Algebra OperationsThe The UNION operationUNION operation


(a) Two union-compatible relations. (b) STUDENT INSTRUCTOR.(c) STUDENT INSTRUCTOR. (d) STUDENT – INSTRUCTOR. (e) INSTRUCTOR – STUDENT

Relational Algebra OperationsRelational Algebra Operations


• Two relations R1 and R2 are said to be union compatible if they have the same degree and all their attributes (correspondingly) have the same domain.

• The UNION, INTERSECTION, and SET DIFFERENCE operations are applicable on union compatible relations

• The resulting relation has the same attribute names as the first relation

Relational Algebra Operations


• The result of UNION operation on two relations, R1 and R2, is a relation, R3, that includes all tuples that are either in R1, or in R2, or in both R1 and R2.

• The UNION operation is denoted by:R3 = R1 R2

• The UNION operation eliminates duplicate tuples

The UNION operation


• The result of INTERSECTION operation on two relations, R1 and R2, is a relation, R3, that includes all tuples that are in both R1 and R2.

• The INTERSECTION operation is denoted by:R3 = R1 R2

• The both UNION and INTERSECTION operations are commutative and associative operations

The INTERSECTION operation


• The result of SET DIFFERENCE operation on two relations, R1 and R2, is a relation, R3, that includes all tuples that are in R1 but not in R2.

• The SET DIFFERENCE operation is denoted by:R3 = R1 – R2

• The SET DIFFERENCE (or MINUS) operation is not commutative

The SET DIFFERENCE Operation


(a) Two union-compatible relations. (b) STUDENT INSTRUCTOR.(c) STUDENT INSTRUCTOR. (d) STUDENT – INSTRUCTOR. (e) INSTRUCTOR – STUDENT

Relational Algebra OperationsRelational Algebra Operations


(a) PROJ_DEPT PROJECT × DEPT. (b) DEPT_LOCS DEPARTMENT × DEPT_LOCATIONS.

The CARTESIAN PRODUCT OperationBinary Relational Operations


• This operation (also known as CROSS PRODUCT or CROSS JOIN) denoted by:

R3 = R1 R2

• The resulting relation, R3, includes all combined tuples from two relations R1 and R2

• Degree (R3) = Degree (R1) + Degree (R2)

• |R3| = |R1| |R2|

The The CARTESIAN PRODUCT CARTESIAN PRODUCT OperationOperation


• An important operation for any relational database is the JOIN operation, because it enables us to combine related tuples from two relations into single tuple

• The JOIN operation is denoted by:R3 = R1 ⋈ <join condition> R2

• The degree of resulting relation is degree(R1) + degree(R2)

Binary Relational Operations


• The difference between CARTESIAN PRODUCT and JOIN is that the resulting relation from JOIN consists only those tuples that satisfy the join condition

• The JOIN operation is equivalent to CARTESIAN PRODUCT and then SELECT operation on the result of CARTESIAN PRODUCT operation, if the select-condition is the same as the join condition

The JOIN Operation


DEPT_MGR DEPARTMENT ⋈ MGRSSN=SSN EMPLOYEE .

JOIN operationJOIN operation


• If the JOIN operation has equality comparison only (that is, = operation), then it is called an EQUIJOIN operation

• In the resulting relation on an EQUIJOIN operation, we always have one or more pairs of attributes that have identical values in every tuples

The EQUIJOIN Operation


• In EQUIJOIN operation, if the two attributes in the join condition have the same name, then in the resulting relation we will have two identical columns. In order to avoid this problem, we define the NATURAL JOIN operation

• The NATURAL JOIN operation is denoted by:R3 = R1 *<attribute list> R2

• In R3 only one of the duplicate attributes from the list are kept

The NATURAL JOIN Operation


ID ENMAE SALARY DNO DNO DNMAE

1001 Ahmed 800 3 3 Operation

1002 Ali 1100 3 3 Operation

1003 Hind 500 1 1 Administration

1004 Omer 1200 1 1 Administration

1005 Salwa 600 3 3 Operation

1006 Issa 1000 1 1 Administration

1007 Noura 500 1 1 Administration

ID ENMAE SALARY DNO

1001 Ahmed 800 3

1002 Ali 1100 3

1003 Hind 500 1

1004 Omer 1200 1

1005 Salwa 600 3

1006 Issa 1000 1

1007 Noura 500 1

Employee:

EMPLOTEE ⋈ DNO=DNO DEPARTMENT

DNO DNMAE1 Administration2 Personal 3 Operation

Department:

*

Variations of JOIN: The EQUIJOIN and NATURAL JOIN (cont’d.)

• 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


• The set of relational algebra operations

{σ, π, , -, x }

• Is a complete set. That is, any of the other relational algebra operations can be expressed as a sequence of operations from this set.

• For example:

R S = (R S) – ((R – S) (S – R))

A Complete Set of Relational Algebra

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

Operations of Relational Algebra

Operations of Relational Algebra (cont’d.)

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

Additional Relational Operations

• 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

Additional Relational Operations (cont’d.)

• Group tuples by the value of some of their attributes – Apply aggregate function independently to

each group

Recursive Closure Operations

• Operation applied to a recursive relationship between tuples of same type

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:

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– Tut tuples with the same value combination

will appear only once

Examples of Queriesin Relational Algebra

Examples of Queriesin Relational Algebra (cont’d.)

Notation for Query Graphs

dr. mohamed hegazi1 the relational algebra and relational calculus

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