+Division

Not supported as a primitive operator, but useful for expressing queries like: Find sailors who have reserved all boats.

Precondition: in A/B, the attributes in B must be included in the schema for A. Also, the result has attributes A-B. SALES(supId, prodId); PRODUCTS(prodId); Relations SALES and PRODUCTS must be built using projections. SALES/PRODUCTS: the ids of the suppliers supplying ALL products.

+Examples of Division A/B

sno pnos1 p1s1 p2s1 p3s1 p4s2 p1s2 p2s3 p2s4 p2s4 p4

pnop2

pnop2p4

pnop1p2p4

snos1s2s3s4

snos1s4

snos1

A

B1B2

B3

A/B1 A/B2 A/B3

+Housekeeping – Jan 25, 2013

+Today:

Quick review of the various relational algebra concepts using small example tables

Monday: work through exercises based on our running example of the transit scheme (I will finish off our first draft of the table scheme today) Goal: see if we can retrieve the information needed to

support our scenarios

+Quiz Confusion (a.k.a. don’t try to remember how you wrote a quiz on less than 4 hours sleep)

Q6. What is the role of a foreign key in a tablea. Links two or more tables togetherb. Uniquely identifies each record in the table

for which it is the foreign keyc. Uniquely identifies each record in the table

from which it originatesd. A and C

+Key selection

A candidate key is any set of one or more columns whose combined values are unique among all occurrences (i.e., tuples or rows). Since a null value is not guaranteed to be unique, no component of a candidate key is allowed to be null. There can be any number of candidate keys in a table

The primary key of any table is any candidate key of that table which the database designer arbitrarily designates as "primary". The primary key may be selected for convenience, comprehension, performance, or any other reasons.

The alternate keys of any table are simply those candidate keys which are not currently selected as the primary key.

A foreign key is a set of one or more columns in any table (not necessarily a candidate key, let alone the primary key, of that table) which may hold the value(s) found in the primary key** column(s) of some other table. ** see slide 7

+Entity Integrity Constraint

No primary key value can be null. Need to be able to identify individual tuples

in the relation – if allowed to have a null value for the primary key, may not be able to distinguish between two or more tuples

+Referential Integrity Constraint Specified between two relations

Used to maintain consistency among tuples of the two relations

A tuple in one relation that refers to another relation must refer to an EXISTING TUPLE in that relation

Foreign keys specify a referential integrity constraint between the two relation A set of attributes FK in relation schema R1 is a foreign key

of R1 if: The attributes in FK have the same domain as the primary

key attributes PK of another relation R2 (FK references R2) A value of FK in a tuple t1 of R1 either occurs as a value of

PK for some tuple T=t2 in R2 or is null. E.g. if dept # is the foreign key of employee, an

employee must either be assigned a value of the primary key in Dept or the value can be Null in employee if no department is currently assigned

+Does a foreign key have to link to a primary key? Slightly controversial. Remember that there could be several candidate

keys from which a single primary key was chosen.

To ensure that each record in the referencing table references exactly one record in the referenced table, the referenced column(s) in the referenced table must have a primary key constraint or have both unique and not-null constraints. Having a unique index is not sufficient.

Under the hood: When you specify a primary key constraint for a table, the Database Engine enforces data uniqueness by automatically creating a unique index for the primary key columns. This index also permits fast access to data when the primary key is used in queries. If a primary key constraint is defined on more than one column, values may be duplicated within one column, but each combination of values from all the columns in the primary key constraint definition must be unique.

More info on foreign key constraints (but remember, this is DB specific): http://msdn.microsoft.com/en-us/library/ms175464(v=sql.105).aspx

+

Relational Algebra CSCI 2141

W 2013

+What is Relational Algebra?

Relational Algebra is formal description of how relational database operates.

It is a procedural query language, i.e. user must define both “how” and “what” to retrieve.

It consists of a set of operators that consume either one or two relations as input. An operator produces one relation as its output.

+Introduction to Relational Algebra

Introduced by E. F.

Codd in 1970.

Codd proposed such

an algebra as a basis

for database query

languages.

+Relational Query Languages

Formal: relational algebra, relational calculus, Datalog

Actual: SQL, Quel, Query-by-Example (QBE)

In ALL languages, a query is executed over a set of relations, get single relation as the result

+Terminology

Relation - a set of tuples.

Tuple - a collection of attributes which describe some real world entity.

Attribute - a real world role played by a named domain.

Domain - a set of atomic values.

Set - a mathematical definition for a collection of objects which contains no duplicates.

+Algebra Operations

Unary Operations - operate on one relation. These include select, project and rename operators.

Binary Operations - operate on pairs of relations. These include union, set difference, division, cartesian product, equality join, natural join, join and semi-join operators.

+Select Operator

The Select operator selects tuples that satisfies a predicate; e.g. retrieve the employees whose salary is 30,000

б Salary = 30,000 (Employee)

Conditions in Selection:

Simple Condition: (attribute)(comparison)(attribute)

(attribute)(comparison)(constant)

Comparison: =,≠,≤,≥,<,>

Condition: combination of simple conditions with AND, OR, NOT

Select Operator Example

Name Age Weight

Harry 34 80

Sally 28 64

George 29 70

Helena 54 54

Peter 34 80

Person бAge≥34(Person)

бAge=Weight(Person)

Select Operator Example

Name Age Weight

Harry 34 80

Sally 28 64

George 29 70

Helena 54 54

Peter 34 80

Name Age Weight

Harry 34 80

Helena 54 54

Peter 34 80

Name Age Weight

Helena 54 54

Person бAge≥34(Person)

бAge=Weight(Person)

+Project Operator

Project (∏) retrieves a column. Duplication is not permitted.

e.g., name of employees:

∏ name(Employee)

e.g., name of employees earning more than 80,000:

∏ name(бSalary>80,000(Employee))

Project Operator Example

Name Age Salary

Harry 34 80,000

Sally 28 90,000

George 29 70,000

Helena 54 54,280

Peter 34 40,000

Employee

∏ name(Employee)

Project Operator Example

Name Age Salary

Harry 34 80,000

Sally 28 90,000

George 29 70,000

Helena 54 54,280

Peter 34 40,000

Name

Harry

Sally

George

Helena

Peter

Employee

∏ name(Employee)

Project Operator Example:∏ name(бSalary>80,000(Employee))

Name Age Salary

Harry 34 80,000

Sally 28 90,000

George 29 70,000

Helena 54 54,280

Peter 34 40,000

Employee бSalary>80,000(Employee)

∏ name(бSalary>80,000(Employee))

Project Operator Example

Name Age Salary

Harry 34 80,000

Sally 28 90,000

George 29 70,000

Helena 54 54,280

Peter 34 40,000Name

Sally

Name Age Salary

Sally 28 90,000

Employee бSalary>80,000(Employee)

∏ name(бSalary>80,000(Employee))

+Cartesian Product

In mathematics, it is a set of all pairs of elements (x, y) that can be constructed from given sets, X and Y, such that x belongs to X and y to Y.

It defines a relation that is the concatenation of every tuple of relation R with every tuple of relation S.

+Cartesian Product

Note:

Relation schema is the union of schemas for R and S

Resulting schema may be ambiguous Use R.A or S.A to disambiguate an attribute that

occurs in both schemas

Cartesian Product Example

Name Age Weight

Harry 34 80

Sally 28 64

George 29 70

City

San Jose

Austin

Person City

Person X City

Cartesian Product Example

Name Age Weight

Harry 34 80

Sally 28 64

George 29 70

City

San Jose

Austin

Name Age Weight City

Harry 34 80 San Jose

Sally 28 64 San Jose

George 29 70 San Jose

Harry 34 80 Austin

Sally 28 64 Austin

George 29 70 Austin

Person City

Person X City

+Example

A B

1 2

3 4

B C

2 5

4 7

D

6

8

9 10 11

x

R S

+Example

A B

1 2

3 4

B C

2 5

4 7

D

6

8

9 10 11

x

A R.BS.B C D

R S

1 2 2 5 6

1 2 4 7 8

1 2 9 10 11

3 4

3 4

3 4

2 5 6

4 7 8

9 10 11

+Rename Operator

In relational algebra, a rename is a unary operation written as ρ a / b (R) where:

a and b are attribute names

R is a relation

The result is identical to R except that the b field in all tuples is renamed to an a field.

Example, rename operator changes the name of its input table to its subscript, ρ EmployeeName / Name (Employee) Changes the Name field of the Employee table to

EmployeeName

Rename Operator Example

Name Salary

Harry 80,000

Sally 90,000

George 70,000

Helena 54,280

Peter 40,000

EmployeeName Salary

Harry 80,000

Sally 90,000

George 70,000

Helena 54,280

Peter 40,000

Employee ρ EmployeeName / Name (Employee)

+Union Operator

The union operation is denoted U as in set theory. It returns the union (set union) of two compatible relations.

For a union operation r U s to be legal, we require that,

r and s must have the same number of attributes.

The domains of the corresponding attributes must be the same.

As in all set operations, duplicates are eliminated.

Union Operator Example

FN LN

Susan Yao

Ramesh Shah

Barbara Jones

Amy Ford

Jimmy Wang

FN LN

John Smith

Ricardo Brown

Susan Yao

Francis Johnson

Ramesh Shah

FN LN

Susan Yao

Ramesh Shah

Barbara Jones

Amy Ford

Jimmy Wang

John Smith

Ricardo Brown

Francis Johnson

StudentProfessor

Student U Professor

+Intersection Operator

Denoted as . For relations R and S, intersection is R S. Defines a relation consisting of the set of all

tuples that are in both R and S. R and S must be union-compatible.

Intersection Operator Example

FN LN

Susan Yao

Ramesh Shah

Barbara Jones

Amy Ford

Jimmy Wang

FN LN

John Smith

Ricardo Brown

Susan Yao

Francis Johnson

Ramesh Shah

FN LN

Susan Yao

Ramesh Shah

StudentProfessor

Student Professor

+Set Difference Operator

For relations R and S,

Set difference R - S, defines a relation consisting of the tuples that are in relation R, but not in S.

Set difference S – R, defines a relation consisting of the tuples that are in relation S, but not in R.

Note: R-S S-R!

Set Difference Operator Example

FN LN

Susan Yao

Ramesh Shah

Barbara Jones

Amy Ford

Jimmy Wang

FN LN

John Smith

Ricardo Brown

Susan Yao

Francis Johnson

Ramesh Shah

Student Professor

Student - ProfessorProfessor - Student

Set Difference Operator Example

FN LN

Susan Yao

Ramesh Shah

Barbara Jones

Amy Ford

Jimmy Wang

FN LN

John Smith

Ricardo Brown

Susan Yao

Francis Johnson

Ramesh Shah

FN LN

Barbara Jones

Amy Ford

Jimmy Wang

Student Professor

Student - ProfessorFN LN

John Smith

Ricardo Brown

Francis Johnson

Professor - Student

+Try R S, R S, R - S

name address gender birthdateCarrie Fisher

Mark Hamil

123 Maple St., Hollywood

456 Oak Rd., Brentwood

F

M

9/9/99

8/8/88

R

name address gender birthdateCarrie Fisher

Harrison Ford

123 Maple St., Hollywood

789 Palm Dr., Beverly Hills

F

M

9/9/99

7/7/77

S

+Sample Operations

name address gender birthdateCarrie Fisher 123 Maple St., Hollywood F 9/9/99

R S

name address gender birthdateCarrie Fisher

Harrison Ford

123 Maple St., Hollywood

789 Palm Dr., Beverly Hills

F

M

9/9/99

7/7/77

R S

Mark Hamil 456 Oak Rd., Brentwood M 8/8/88

name address gender birthdateR - S

Mark Hamil 456 Oak Rd., Brentwood M 8/8/88

+Natural Join Operator

Natural join is a dyadic operator that is written as R lXl S where R and S are relations. The result of the natural join is the set of all combinations of tuples in R and S that are equal on their common attribute names.

If r and s from r(R) and s(S) are successfully paired, result is called a joined tuple

Natural Join Example

Name EmpID DeptName

Harry 3415 Finance

Sally 2241 Sales

George 3401 Finance

Harriet 2202 Sales

DeptName Mgr

Finance George

Sales Harriet

Production Charles

Name EmpID DeptName Mgr

Harry 3415 Finance George

Sally 2241 Sales Harriet

George 3401 Finance George

Harriet 2202 Sales Harriet

Employee

Dept Employee lXl Dept

For an example, consider the tables Employee and Dept and their natural join:

+Example

A B

1 2

3 4

B C

2 5

4 7

D

6

8

9 10 11

join

Resulting schema has attributes from R, either R or S (i.e., joining attribute(s)), and S

Tuples that fail to pair with any tuple of the other relation are called dangling tuples

R S

+Example

A B

1 2

3 4

B C

2 5

4 7

D

6

8

9 10 11

join

A B C D

R S

1 2 5 6

3 4 7 8

+Join Operations

Theta Join (binary)

R joinC S, where C is an arbitrary join condition

Step 1: take the product of R and S

Step 2: Select from the product only those tuples that satisfy condition C

As with the product operation, the schema for the result is the union of the schemas of R and S

+Example

B C

2 3

2 3

D

4

5

7 8 10

joinA<D AND U.BV.B

A B

1 2

6 7

C

3

8

9 7 8

U V

+Example

B C

2 3

2 3

D

4

5

7 8 10

joinA<D AND U.BV.B

A B

1 2

6 7

C

3

8

9 7 8

U V

V.BV.C DA U.BU.C

1 2 3 7 8 10

+Additional slides w/ other types of joinsFor more information of the various relational algebra operations, see: http://en.wikipedia.org/wiki/Relational_algebra

+Semijoin Operator

The semijoin is joining similar to the natural join and written as R ⋉ S where R and S are relations. The result of the semijoin is only the set of all tuples in R for which there is a tuple in S that is equal on their common attribute names.

Semijoin Example

Name EmpID DeptName

Harry 3415 Finance

Sally 2241 Sales

George 3401 Finance

Harriet 2202 Sales

DeptName Mgr

Sales Harriet

Production Charles

Name EmpID DeptName

Sally 2241 Sales

Harriet 2202 Sales

Employee

DeptEmployee ⋉ Dept

For an example consider the tables Employee and Dept and their semi join:

+Outerjoin Operator

Left outer joinThe left outer join is written as R =X S where R and S are relations. The result of the left outer join is the set of all combinations of tuples in R and S that are equal on their common attribute names, in addition to tuples in R that have no matching tuples in S.

Right outer joinThe right outer join is written as R X= S where R and S are relations. The result of the right outer join is the set of all combinations of tuples in R and S that are equal on their common attribute names, in addition to tuples in S that have no matching tuples in R.

Left Outerjoin Example

Name EmpID DeptName

Harry 3415 Finance

Sally 2241 Sales

George 3401 Finance

Harriet 2202 Sales

DeptName Mgr

Sales Harriet

Name EmpID DeptName Mgr

Harry 3415 Finance ω

Sally 2241 Sales Harriet

George 3401 Finance ω

Harriet 2202 Sales Harriet

Employee

Dept

Employee =X Dept

For an example consider the tables Employee and Dept and their left outer join:

Right Outerjoin Example

Name EmpID DeptName

Harry 3415 Finance

Sally 2241 Sales

George 3401 Finance

Harriet 2202 Sales

DeptName Mgr

Sales Harriet

Production Charles

Name EmpID DeptName Mgr

Sally 2241 Sales Harriet

Harriet 2202 Sales Harriet

ω ω Production Charles

Employee

Dept

Employee X= Dept

For an example consider the tables Employee and Dept and their right outer join:

Full Outer join Example

Name EmpID DeptName

Harry 3415 Finance

Sally 2241 Sales

George 3401 Finance

Harriet 2202 Sales

DeptName Mgr

Sales Harriet

Production Charles

Name EmpID DeptName Mgr

Harry 3415 Finance ω

Sally 2241 Sales Harriet

George 3401 Finance ω

Harriet 2202 Sales Harriet

ω ω Production Charles

Employee

Dept

Employee =X= Dept

The outer join or full outer join in effect combines the results of the left and right outer joins.For an example consider the tables Employee and Dept and their full outer join:

+References

http://en.wikipedia.org/wiki/Relational_algebra#Outer_join

http://www.cs.sjsu.edu/faculty/lee/cs157/cs157alecturenotes.htm

Database System Concepts, 5th edition, Silberschatz, Korth, Sudarshan