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Database Systems
DataBase System
Major Content & Grade
 Introduction
*
 The
***
Relational Model
 SQL
****
 Transaction
Management
***
 Database
Design (E-R)
***
 Database
Design (Normalization)
***
Haichang Gao , Software School , Xidian University
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DataBase System
Unit 2 The Relational Model
 Relational
Model
 Relational
Algebra(关系代数)
 Relational
Calculus(关系演算)
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DataBase System
Relational Model
 Laid
down in 1969-1970 by E.F.CODD
“ A Relational Model of Data for Large Shared
Data Bank” C ACM 1970

Mathematical Bases ( Relational Theory)

Developed in 1980s

Most commercial DBMS are Relational
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DataBase System
Mathematical Relations
given sets D1, D2, …. Dn, a relation r is a subset of
Cartesian product(笛卡尔积) D1 × D2 × … × Dn
Thus a relation(关系) is a set of n-tuples (a1, a2, …, an)
where ai  Di
 Example: if
customer-name = {Jones, Smith, Curry, Lindsay}
customer-street = {Main, North, Park}
customer-city = {Harrison, Rye, Pittsfield}
Then r = {(Jones, Main, Harrison),
(Smith, North, Rye),
(Curry, North, Rye),
(Lindsay, Park, Pittsfield)}
is a relation over customer-name × customer-street ×
customer-city.
 Formally,
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DataBase System
Mathematical Relations
 Each attribute of a relation has a name
 The set of allowed values for each attribute is called the
domain（域） of the attribute
 Attribute values are (normally) required to be atomic; that is,
indivisible
 Note:
multivalued attribute(多值属性) values are not atomic
 Note:
composite attribute (组合属性) values are not atomic
 The special value null is a member of every domain
 The
null value causes complications in the definition of many
operations
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DataBase System
Mathematical Relations
 A1,
R
A2, …, An are attributes
= (A1, A2, …, An ) is a relation schema(关系模式)
E.g. Customer-schema =
(customer-name, customer-street, customer-city)
 r(R)
is a relation on the relation schema R
E.g.
customer (Customer-schema)
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DataBase System
Mathematical Relations
current values (relation instance, 关系实例) of a relation
are specified by a table
 An element t of r is a tuple(元组), represented by a row in a
table
 Order of tuples is irrelevant (元组是无序的 tuples may be
stored in an arbitrary order)
 The
attributes
(or columns)
customer_namecustomer_street customer_city
Jones
Smith
Curry
Lindsay
Main
North
North
Park
Harrison
Rye
Rye
Pittsfield
tuples
(or rows)
customer
Haichang Gao , Software School , Xidian University
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DataBase System
Relational Database
A
Relational database consists of multiple relations
 Information about an enterprise is broken up into parts, with
each relation storing one part of the information
 Example:
The customer table stores information about customers
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DataBase System
Relational Database
(b) The account table
(c) The depositor table
The account table stores information about accounts.
The depositor table containing information about which
customer owns which account
 Storing all information as a single relation is NOT a good
idea!
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DataBase System
Key
 Let K  R
 K is a superkey(超码) of R if values for K are sufficient to
identify a unique tuple of each possible relation r(R).
 Example:
{customer_id, customer_name} and
{customer_id} are both superkeys of Customer.
 by “possible r” we mean a relation r that could exist in the
enterprise we are modeling.
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DataBase System
Key
 Let K  R
 K is a superkey(超码) of R if values for K are sufficient to
identify a unique tuple of each possible relation r(R).
 K is a candidate key(候选码) if K is minimal.
 Example:
{customer_id} is a candidate key for Customer,
since it is a superkey, and no subset of it is a superkey.
 {customer_name} is a candidate key also, assuming no two
customers can possibly have the same name.
Haichang Gao , Software School , Xidian University
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DataBase System
Key
 Primary key(主码): The candidate key that is selected to
identify tuples uniquely within the relation. And the others
candidate key is called Alternate Keys(替换码).
 Example: for relation
customer (customer_id, customer_name, customer_street,
customer_city)
 Candidate key : {customer_id}, {customer_name}
 Primary key: {customer_id}
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DataBase System
Key
 For the banking database, we have several relations to store data:
 Foreign Key(外码): An attribute, or set of attributes,within one
relation R1 that matches the candidate key of some(possibly the
same) relation R2.
 Example: Attribute customer_name
in table depositor is a
Foreign Key references to customer_name in customer.
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DataBase System
Key
 Usually, Primary Key and Foreign Key was indicated using
undirected line and dashed line.
 Example:
Find out PK and FK of the following schema:
Student(sno, sname, ssex, sbirth, sdept)
Course(cno, cname, cpno, ccredit)
SC(sno, cno, grade)
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DataBase System
Integrity of Relational Model
 Entity integrity(实体完整性约束): In a base relation, NO
attribute of a primary key can be null.
 Referential integrity(参照完整性约束): If a foreign key exists
in a relation, either the foreign key value must match a candidate
key value of some tuple in its home relation(被参照关系) or the
foreign key value must be wholly null.
 Example:
customer (customer_name, customer_street, customer_city)
depositor (customer_name, account_number)
account (account_number, branch_name, balance)
Haichang Gao , Software School , Xidian University
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DataBase System
Integrity of Relational Model
 Null(空值): Represents a value for an attribute that is
currently unknown or is not applicable for this tuple.
 It
is possible for tuples to have a null value, denoted by null,
for some of their attributes
 The
result of any arithmetic expression involving null is null.
 Aggregate functions
simply ignore null values (as in SQL)
 For
duplicate elimination and grouping, null is treated like
any other value, and two nulls are assumed to be the same
(as in SQL)
 Null
value is NOT 0 of integer, NOT empty string , it is
JUST NULL.
Haichang Gao , Software School , Xidian University
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DataBase System
Integrity of Relational Model
 Comparisons with null values return the special truth value:
unknown
 If false was used instead of unknown, then not (A < 5)
would not be equivalent to
A >= 5
 Three-valued logic using the truth value unknown:
 OR: (unknown or true)
= true,
(unknown or false)
= unknown
(unknown or unknown) = unknown
 AND: (true and unknown)
= unknown,
(false and unknown)
= false,
(unknown and unknown) = unknown
 NOT: (not unknown) = unknown
 Result of select predicate is treated as false if it evaluates to
unknown
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DataBase System
Unit 2 The Relational Model
 Relational
Model
 Relational
Algebra(关系代数)
 Relational
Calculus(关系演算)
Haichang Gao , Software School , Xidian University
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DataBase System
Manipulation of Relational Model
 Query Language is a Language in which user requests
information from the database.
 Categories of languages
 Procedural(过程化的)
 Non-procedural, or declarative(非过程化的，声明性的)
 Mathematical languages:
 Relational algebra(关系代数)
 Tuple relational calculus(元组演算)
 Domain relational calculus (域演算)
 Characteristics of Relational languages
 all set-at-a-time (一次一集合)
operands & results are whole tables
 Closure property（封闭性）
the output from one operation can become input to another.
Haichang Gao , Software School , Xidian University
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DataBase System
Relational Algebra
 Relational algebra is a Procedural language
 Basic operators
 select(选择): 
 project(投影): π
 union(并):
 set
∪
difference(集合差): –
 Cartesian product(笛卡尔积): ×
 rename:

 The operators take two or more relations as inputs and give a
new relation as a result.
Haichang Gao , Software School , Xidian University
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DataBase System
Basic operators
 Select Operation(选择操作)
 Notation:
 p(r)
p
is called the selection predicate(条件谓词)
 Defined as:
p(r) = {t | t  r  p(t)}
Where p is a formula in propositional calculus (命题演算)
consisting of terms connected by :  (and),  (or),  (not)
Each term is one of:
<attribute> op <attribute> (or <constant>)
where op is one of: =, , >, , <,  (comparison operator)
 Example of selection:
 branch_name=“Perryridge”(account)
Haichang Gao , Software School , Xidian University
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DataBase System
Basic operators
 Select Operation(选择操作)
• Relation r
• A=B  D > ‘5’ (r)
A
B
C
D


1
7


5
7


12
3


23 10
A
B
C
D


1
7


23 10
Haichang Gao , Software School , Xidian University
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DataBase System
Basic operators
 Project Operation(投影操作)
 Notation:
π A1, A2, …, Ak (r)
where A1, A2 are attribute names and r is a relation name.
 The
result is defined as the relation of k columns obtained by
erasing the columns that are not listed
 Duplicate rows
 Example:
removed from result, since relations are sets
For relation
account (account_number, branch_name, balance)
To eliminate the branch_name attribute of account
π account_number, balance (account)
Haichang Gao , Software School , Xidian University
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DataBase System
Basic operators
 Project Operation(投影操作)
Relation r:
π A,C (r)
A
B
C

10
1

20
1

30
1

40
2
A
C
A
C

1

1

1

1

1

2

2
=
Haichang Gao , Software School , Xidian University
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DataBase System
Basic operators
 Union Operation(并操作)
 Notation:
 Defined
rs
r union s
as:
r  s = {t | t  r  t  s}
 For
r  s to be valid:
1. r, s must have the same degree(列数相同)
2. The attribute domains must be compatible (域相容)

Example: find all customers with either an account or a loan
π customer_name (depositor)  π customer_name (borrower)
Haichang Gao , Software School , Xidian University
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DataBase System
Basic operators
 Union Operation(并操作)
Relations r, s:
A
B
A
B

1

2

2

3

1
s
r
r  s:
A
B

1

2

1

3
The 1st column of r deals with
the same type of values(domain)
as does the 1st column of s ,and
2nd column also, and also for all
attributes.
Haichang Gao , Software School , Xidian University
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DataBase System
Basic operators
 Difference Operation(差操作)
 Notation:
 Defined
r–s
r minus s
as:
r – s = {t | t  r  t  s}
 Set
differences must be taken between compatible relations.

r and s must have the same degree

attribute domains of r and s must be compatible
Haichang Gao , Software School , Xidian University
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DataBase System
Basic operators
 Difference Operation (差操作)
Relations r, s:
A
B
A
B

1

2

2

3

1
s
r
r–s:
A
B

1

1
Haichang Gao , Software School , Xidian University
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DataBase System
Basic operators
 Cartesian-Product Operation(笛卡尔积操作)
 Notation:
 Defined
r×s
r times s
as:
r × s = {tq | t  r  q  s}
 Assume
that attributes of r(R) and s(S) are disjoint. (That is,
R  S =  , NO same attribute in R and S).
 If
attributes of r(R) and s(S) are not disjoint, then renaming
must be used.
Haichang Gao , Software School , Xidian University
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DataBase System
Basic operators
 Cartesian-Product Operation(笛卡尔积操作)
Relations r, s:
A
B
C
D
E

1

2




10
10
20
10
a
a
b
b
r
s
r × s:
A
B
C
D
E








1
1
1
1
2
2
2
2








10
19
20
10
10
10
20
10
a
a
b
b
a
a
b
b
Haichang Gao , Software School , Xidian University
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DataBase System
Basic operators
 Rename Operation(重命名操作)
 Allows
us to name, and therefore to refer to, the results of
relational-algebra expressions.
 Allows us to refer to a relation by more than one name.
 Example:
 x (E)
returns the expression E under the name x
 If a relational-algebra expression E has degree n, then
x (A1, A2, …, An) (E)
A1, A2, …., An returns the result of expression E under
the name x, and with the attributes renamed to A1, A2, ….,
An.
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Example Queries
 Find the largest account balance.
DataBase System
account_
number
Branch_
name
A-101
A-215
A-102
…
Downtown
Mianus
Perryridge
……
balance
500
700
400
…
account
 A1.balance >A2.balance (A1(account) × A2(account) )
?
最小值不被选中!
πbalance(account) –
πA1.balance (A1.balance < A2.balance (A1(account) × A2(account) ) )
33
Haichang Gao , Software School , Xidian University
Database Systems
DataBase System
Unit 2 The Relational Model
 Relational
Model
 Relational
Algebra(关系代数)
 Relational
Calculus(关系演算)
Haichang Gao , Software School , Xidian University
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DataBase System
Relational Algebra expression
 A basic expression in the relational algebra consists of either
one of the following:
 A relation in the database
 A constant relation
 Let E1 and E2 be relational-algebra expressions; the
following are all relational-algebra expressions:
 E1  E2
 E1 – E2
 E1 × E2
 p(E1), P is a predicate on attributes in E1
 πs(E1), S is a list consisting of some of the attributes in E1
 x(E1), x is the new name for the result of E1
Haichang Gao , Software School , Xidian University
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DataBase System
Example Queries
 Banking DataBase:
branch (branch_name, branch_city, assets)
customer (customer_name, customer_street, customer_city)
depositor (customer_name, account_number)
account (account_number, branch_name, balance)
borrower (customer_name, loan_number)
loan (loan_number, branch_name, amount)
Haichang Gao , Software School , Xidian University
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DataBase System
Example Queries
 Find the loan number for each loan of an amount greater than
$1200.
loan_number branch_name amount
L-11
L-14
L-15
L-16
L-17
L-23
L-93
Round Hill
Downtown
Perryridge
Perryridge
Downtown
Redwood
Mianus
loan
900
1500
1500
1300
1000
2000
500
πloan_number ( amount > ‘1200’ (loan) )
loan_number
L-14
L-15
L-16
L-23
注：除特别要求外，写查询表
达式时不需要写出结果集！
Haichang Gao , Software School , Xidian University
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DataBase System
Example Queries
account_
number
A-101
A-215
A-102
A-305
A-201
A-222
A-217
branch_
name
Downtown
Mianus
Perryridge
Round Hill
Brighton
Redwood
Brighton
account
balance
500
700
400
350
900
700
750
customer_
account_
name
number
Hayes
A-102
Johnson
A-101
Johnson
A-201
Jones
A-217
Lindsay
A-222
Smith
A-215
Turner
A-305
depositor
 Find the customers name who have at least one deposit of a
balance greater than $700.
πCustomer_name(account.account_number = depositor.account_number (
π account_number(balance>’700’(account))×depositor ) )
πCustomer_name(account.account_number = depositor.account_number  balance>’700’ (
A1:
A2:
account ×depositor ) )
Haichang Gao , Software School , Xidian University
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DataBase System
Example Queries
customer_
account_
name
number
Hayes
A-102
Johnson
A-101
Johnson
A-201
Jones
A-217
Lindsay
A-222
Smith
A-215
Turner
A-305
depositor
D1.cname D1.a# D2.cname D2.a#
Hayes
Hayes
Hayes
……
Johnson
……
Johnson
……
A-102
A-102
A-102
……
A-101
……
A-201
……
Hayes
Johnson
Johnson
……
Johnson
……
Johnson
……
A-102
A-101
A-201
……
A-201
……
A-101
……
 Find all customers who have at least two deposits.
πD1.cname(D1.cname=D2.cname ∧ D1.a#<>D2.a# (
D1(cname,a#)(depositor) × D2(cname,a#)( depositor) ) )
Haichang Gao , Software School , Xidian University
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Example Queries
Branch_
name
A-101
A-215
A-102
…
Downtown
Mianus
Perryridge
……
 Find the largest account balance.
A.ano A.bn
A-101
A-101
A-101
A-215
A-215
A-215
A-102
A-102
A-102
…
Downtown
Downtown
Downtown
Mianus
Mianus
Mianus
Perryridge
Perryridge
Perryridge
…….
A.b B.ano B.bn
500
500
500
700
700
700
400
400
400
…
A-101
A-215
A-102
A-101
A-215
A-102
A-101
A-215
A-102
…
B.b
Downtown
Mianus
Perryridge
Downtown
Mianus
Perryridge
Downtown
Mianus
Perryridge
……
DataBase System
account_
number
500
700
400
500
700
400
500
700
400
…
balance
500
700
400
…
account
A.b > B.b (A × B)?
A.b < B.b (A × B)?
πbalance(account) –
πA1.balance (A1.balance < A2.balance (A1(account) × A2(account) ) )
Haichang Gao , Software School , Xidian University
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DataBase System
Additional Operations
 We define additional operations that do not add any
power to the relational algebra, but that simplify
common queries.
 Set
intersection(集合交)
 Natural
join(自然连接)
 Division(除法)
Haichang Gao , Software School , Xidian University
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DataBase System
Additional Operations
 Intersection Operation(交操作)
 Notation: r
s
 Assume:

r, s have the same degree

attributes of r and s are compatible
 Defined
as:
r  s ={ t | t  r  t  s }
 Note:
r  s = r - (r - s)
 Find the names of all customers who have a loan and an account
at bank.
πcustomer_name (borrower )  πcustomer_name (depositor )
Haichang Gao , Software School , Xidian University
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DataBase System
Additional Operations
 -Join Operation(-连接)
 -Join
is a general join which don’t need there are same
attribute for both operators, and the join condition can be any
compare operations. That is:
R times S WHERE F
R
AθB
S = { tr ts | tr  R  ts S  tr[A]  ts[B] }
 Equi-join is another join. Its join condition is equal operator.
R
A=B
S = { tr ts | tr  R  ts S  tr[A]= ts[B] }
Haichang Gao , Software School , Xidian University
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DataBase System
Additional Operations
 -Join Operation(-连接) R
Relations r, s:
R
A
a1
a1
a2
a2
B
b1
b2
b3
b4
S
B
b1
b2
b3
b3
b5
E
3
7
10
2
2
S
C<E
A
a1
R.B
b1
C
5
S.B
b2
E
7
a1
b1
5
b3
10
a1
b2
6
b2
7
a1
b2
6
b3
10
a2
b3
8
b3
10
R
S
C
5
6
8
12
B=B
A
R.B
C
S.B
E
a1
b1
5
b1
3
a1
b2
6
b2
7
a2
b3
8
b3
10
a2
b3
8
b3
2
Haichang Gao , Software School , Xidian University
45
DataBase System
Additional Operations
 Natural join Operation(自然连接)
 Notation:
r s
 Let r and s be relations on schemas R and S respectively.The
result is a relation on schema R  S which is obtained by
considering each pair of tuples tr from r and ts from s.
 If tr and ts have the same value on each of the attributes in
RS, a tuple t is added to the result, where
 t has the same value as tr on r
 t has the same value as ts on s
 Example:
R = (A, B, C, D)
S = (E, B, D)
 Result schema = (A, B, C, D, E)
r
s is defined as:
π r.A, r.B, r.C, r.D, s.E (r.B = s.B r.D = s.D (r × s))
Haichang Gao , Software School , Xidian University
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DataBase System
Additional Operations
 Natural join Operation(自然连接)
Relations r, s:
A
B
C
D
B
D
E





1
2
4
1
2





a
a
b
a
b
1
3
1
2
3
a
a
a
b
b





r
r
s
s
A
B
C
D
E





1
1
1
1
2





a
a
a
a
b





Haichang Gao , Software School , Xidian University
47
DataBase System
Example Queries
Additional Operations simplify the relational algebra
expressions !
 Find the customers name who have at least one deposit of a
balance greater than $700.
A1:
πCustomer_name(account.account_number = depositor.account_number (
π account_number(balance>’700’(account)) ×depositor ) )
A2:
πCustomer_name(account.account_number = depositor.account_number  balance>’700’ (
account ×depositor ) )
A3:
πCustomer_name(balance>’700’ ( account
depositor ) )
Haichang Gao , Software School , Xidian University
48
DataBase System
Example Queries
customer_
account_
name
number
Hayes
A-102
Johnson
A-101
Johnson
A-201
Jones
A-217
Lindsay
A-222
Smith
A-215
Turner
A-305
depositor
account_
number
A-101
A-215
A-102
A-305
A-201
A-222
A-217
branch_
name
Downtown
Mianus
Perryridge
Round Hill
Brighton
Redwood
Brighton
account
balance
500
700
400
350
900
700
750
 (exercise)Find all customers who have at least two deposits in
different branch.
πD1.Customer_name( D1.Customer_name =D2. Customer_name 
D1. Account_number<>D2. Account_number  D1.branch_name<>D2. branch_name(
D1(depositor
account) × D2(depositor
account) )
Haichang Gao , Software School , Xidian University
49
DataBase System
Example Queries
customer_
account_
name
number
Hayes
A-102
Johnson
A-101
Johnson
A-201
Jones
A-217
Lindsay
A-222
Smith
A-215
Turner
A-305
depositor
account_
number
A-101
A-215
A-102
A-305
A-201
A-222
A-217
branch_
name
Downtown
Mianus
Perryridge
Round Hill
Brighton
Redwood
Brighton
account
balance
500
700
400
350
900
700
750
 (exercise)Find all customers who have at least two deposits in
different branch.
πD1.cname(D1.cname=D2.cname  D1.a#<>D2.a#  D1.bname<>D2.bname(
D2(cname,a#,bname)(πdepositor.customer_name,account_number,branch_name
depositor.account_number=account.account_number(depositor × account)) ×
D1(cname,a#,bname)(πdepositor.customer_name,account_number,branch_name
depositor.account_number=account.account_number(depositor × account)) )
Haichang Gao , Software School , Xidian University
50
DataBase System
Additional Operations
 Division Operation(除)
 Suited
to queries that include the phrase “for all”.
 Let
r and s be relations on schemas R and S respectively
where

R = (A1, …, Am, B1, …, Bn)

S = (B1, …, Bn)
The result of r  s is a relation on schema
R  S = (A1, …, Am)
 Defined
as:
r  s = { t | t  π R-S(r)   u  s → tu  r ) }
Haichang Gao , Software School , Xidian University
51
DataBase System
Additional Operations
 Division Operation(除)
Relations r, s:
A
B
B











1
2
3
1
1
1
3
4
6
1
2
1
r  s:
2
s
A


r
Haichang Gao , Software School , Xidian University
52
DataBase System
Additional Operations
 Division Operation(除)
 Another
Defined as:
R  S  {tr [ X ] | tr  R  Yx   Y (S )}
Yx={y | tr∈R  y=tr[Y]  x=tr[X]}
Relations r, s:
r
A B








a
a
a
a
a
a
a
a
C
D
E








a
a
b
a
b
a
b
b
1
1
1
1
3
1
1
1
s
D
E
a
b
1
1
r  s:
A
B
C


a
a


Haichang Gao , Software School , Xidian University
53
DataBase System
Additional Operations
 Division Operation(除)
 Property
Let q is the result of r  s
 Then q is the largest relation satisfying
q×sr
 Definition in terms of the basic algebra operation

Let r(R) and s(S) be relations, and let S  R
r  s = π R-S (r) –πR-S ( (π R-S (r) × s) – πR-S,S(r))
Haichang Gao , Software School , Xidian University
54
DataBase System
Example Queries
 Find all customers who have an account at all branches
located in Brooklyn city.
customer_
name
Johnson
Smith
Hayes
Turner
Johnson
Lindsay
Jones
account_
number
A-101
A-215
A-102
A-305
A-201
A-222
A-217
branch_
balance
name
Downtown
500

Mianus
700
Perryridge
400
Round Hill
350
Brighton
900
Redwood
700
Brighton
750
account
branch_name
Brighton
Downtown
Mianus
North Town
Perryridge
Pownal
Redwood
Round Hill
branch_city
Brooklyn
Brooklyn
Horseneck
Rye
Horseneck
Bennington
PaloAlto
Horseneck
branch
assets
7100000
9000000
400000
3700000
1700000
300000
2100000
8000000
π customer_name, branch_name (depositor
account) 
πbranch_name (branch_city = “Brooklyn” (branch))
Haichang Gao , Software School , Xidian University
55
DataBase System
Extended Operations
 Outer join Operation(外连接)
 The
(left) Outer join is a join on which tuples from R that do
not have matching values in the common attributes of S are
alse included in the result relation. Missing values in the
second relation are set to null.
 Notation:
r
s (left outer join )
A
C
D
A
B




10
10
20
10
a
a
b
b

1

2
r
s
A
C
D
B




10
10
20
10
a
a
b
b
1
2
2
r
s
Haichang Gao , Software School , Xidian University
56
DataBase System
Extended Operations
loan_number
L-170
L-230
L-260
customer_name
loan_number
branch_name amount
L-170
Downtown
3000 Jones
L-230
Redwood
4000 Smith
L-155
Perryridge
1700 Hayes
loan
borrower
 Inner Join: loan
borrower
loan_number branch_Name amount customer_name
L-170
Downtown
3000 Jones
L-230
Redwood
4000 Smith
 Left Outer Join
loan_number
: loan
borrower
branch_Name
amount customer_name
L-170
Downtown
3000 Jones
L-230
Redwood
4000 Smith
L-260
Perryridge
1700 NULL
Haichang Gao , Software School , Xidian University
57
DataBase System
Extended Operations
loan_number
L-170
L-230
L-260
customer_name
Loan_number
branch_name amount
L-170
Downtown
3000 Jones
L-230
Redwood
4000 Smith
L-155
Perryridge
1700 Hayes
loan
borrower
 Right Outer Join
: loan
borrower
loan_number nranch_name
L-170
Downtown
L-230
Redwood
L-155
 Full Outer Join
: loan
loan_number branch_name
L-170
Downtown
L-230
Redwood
L-260
Perryridge
L-155
amount customer_name
3000 Jones
4000 Smith
Hayes
borrower
amount customer_name
3000 Jones
4000 Smith
1700
Hayes
Haichang Gao , Software School , Xidian University
58
DataBase System
Homework 1
 2.1
 2.5
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59
Database Systems
DataBase System
Unit 2 The Relational Model
 Overview
 Relational
Model
 Relational
Algebra(关系代数)
 Relational
Calculus(关系演算)
(教材5.1节)
Haichang Gao , Software School , Xidian University
61
DataBase System
Relational calculus
 Relational Calculus Based on a branch of mathematical
logical called the predicate calculus(谓词演算).
 A nonprocedural query language, where each query is of the
form :
 It
{t | P(t) }
is the set of all tuples t such that predicate P is true for t
t
is a tuple variable, t[A] denotes the value of tuple t on
attribute A, called component of t on attribute A
 R(t)
P
denotes that tuple t is in relation R
is a formula of the predicate calculus
Haichang Gao , Software School , Xidian University
62
DataBase System
Relational calculus
 Tuple-relational-Calculus Formula
 R(t)
is an atom formula(原子公式)
 <t[A] θ t[B]> (or <t[A] θ c> or <c θ t[A] > ) is an atom
formula
θ is one of the comparison operators: (, , , , , )
 if P1, P2 are formulae, then
 P1, P1  P2, P1  P2, P1  P2 are formulae
 x(P(x)), x (P(x)) are formulae
 Sets of equivalent expressions:
 P1  P2  (P1  P2 )
 P1  P2   P1  P2
 t ( P(t) )  t ( P(t) )
Haichang Gao , Software School , Xidian University
63
DataBase System
Relational calculus
 Example
R
S
 R1={ t | S(t)  t[1]> ‘2’}
A
B
C
A
B
C
1
2
3
1
2
3
A
B
C
4
5
6
3
4
6
3
4
6
7
8
9
5
6
9
5
6
9
 R2={ t | R(t) S(t)}
A
B
C
4
5
6
7
8
9
 R3={ t | S(t)  u(R(u)  t[3]<u[2]) }
A
B
C
1
2
3
3
4
6
Haichang Gao , Software School , Xidian University
64
DataBase System
Relational calculus
 Example
R
S
 R4={ t | R(t) u(S(u)t[3]>u[1]) }
A
B
C
A
B
C
1
2
3
1
2
3
4
5
6
3
4
6
7
8
9
5
6
9
A
B
C
4
5
6
7
8
9
Haichang Gao , Software School , Xidian University
65
DataBase System
Relational calculus
 Example
R
S
 R4={ t | R(t) u(S(u)t[3]>u[1]) }
A
B
C
A
B
C
1
2
3
1
2
3
4
5
6
3
4
6
7
8
9
5
6
9
A
B
C
4
5
6
7
8
9
 R5={ t | u v( R(u)  S(v)  u[1]>v[2]  t[1]=u[2] 
t[2]=v[3]  t[3]=u[1] )}
Process：
1. Ensure the schema of result；
2. Obtain tuples according the
predicate calculus .
R.B
S.C R.A
5
8
3
3
4
7
8
6
7
8
9
7
Haichang Gao , Software School , Xidian University
66
DataBase System
Example Queries
 Banking DataBase:
branch (branch_name, branch_city, assets)
customer (customer_name, customer_street, customer_city)
depositor (customer_name, account_number)
account (account_number, branch_name, balance)
borrower (customer_name, loan_number)
loan (loan_number, branch_name, amount)
Haichang Gao , Software School , Xidian University
67
DataBase System
Example Queries
 Find the loan_number, branch_name, and amount for loans of
over $1200.
loan_number branch_name amount
L-11
L-14
L-15
L-16
L-17
L-23
L-93
Round Hill
Downtown
Perryridge
Perryridge
Downtown
Redwood
Mianus
loan
900
1500
1500
1300
1000
2000
500
{t | loan(t)  t[amount]  1200}
 Find the loan number for each loan of an amount greater than
$1200.
{t |  s( loan(s)  s[amount ]  1200 
t [loan_number] = s [loan_number ] )}
Haichang Gao , Software School , Xidian University
68
DataBase System
Example Queries
customer_
account_
name
number
Hayes
A-102
Johnson
A-101
Johnson
A-201
Jones
A-217
Lindsay
A-222
Smith
A-215
Turner
A-305
depositor
account_
number
A-101
A-215
A-102
A-305
A-201
A-222
A-217
branch_
name
Downtown
Mianus
Perryridge
Round Hill
Brighton
Redwood
Brighton
account
balance
500
700
400
350
900
700
750
 Find the customers name who have at least one deposit of a
balance greater than $700.
{t | u( depositor(u)  v( account(v)  u[account_number ]
= v[account_number]  v[balance ] > ‘700’ ) 
t[customer_name] = u[customer_name ] ) }
Haichang Gao , Software School , Xidian University
69
DataBase System
Example Queries
u
customer_
account_
name
number
Hayes
A-102
Johnson
A-101
Johnson
A-201
Jones
A-217
Lindsay
A-222
Smith
A-215
Turner
A-305
depositor
v
customer_
account_
name
number
Hayes
A-102
Johnson
A-101
Johnson
A-201
Jones
A-217
Lindsay
A-222
Smith
A-215
Turner
A-305
depositor
 Find all customers who have at least two deposits.
{t | u v( depositor(u)  depositor(v)  u[customer_name ] =
v[customer_name]  u[account_number]  v [account_number]
 t[customer_name] = u [customer_name ] ) }
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70
DataBase System
x( P(x)  Q(x) )
Example Queries
 x(P(x)  Q(x) )
 Find the largest account balance.
u
account_
number
A-101
A-215
A-102
…
branch_
name
Downtown
Mianus
Perryridge
……
account
A1:
balance
500
700
400
…
v
account_
number
branch_
name
A-101
A-215
A-102
…
Downtown
Mianus
Perryridge
……
balance
500
700
400
…
account
{t | u ( account(u)  v( account(v)  u[balance] ≥
v[balance])  t[balance] = u [balance] ) }
A2:
{t | u ( account(u)    v( account(v)  u[balance] 
v[balance])  t[balance] = u [balance] ) }
Haichang Gao , Software School , Xidian University
71
customer_
account_
name
number
Hayes
A-102
Johnson
A-101
Johnson
A-201
Jones
A-217
Lindsay
A-222
Smith
A-215
Turner
A-305
depositor
account_
number
A-101
A-215
A-102
A-305
A-201
A-222
A-217
Example Queries
u
v
x
y
branch_
name
Downtown
Mianus
Perryridge
Round Hill
Brighton
Redwood
Brighton
account
DataBase System
balance
500
700
400
350
900
700
750
 (exercise)Find all customers who have at least two deposits in
different branch.
{t | u x ( depositor(u)  account(x)  u[account_number] =
x[account_number]  v y ( depositor(v)  account(y) 
v[account_number] = y[account_number]  u[customer_name] =
v[customer_name]  x[branch_name]  y[branch_name] 
u[account_number]  v[account_number] ) t[customer_name] =
u [customer_name] ) }
Haichang Gao , Software School , Xidian University
72
DataBase System
Example Queries
 Find all customers who have an account at all branches
located in Brooklyn city.
customer_ account_
name
number
Hayes
A-102
Johnson
A-101
Johnson
A-201
Jones
A-217
Lindsay
A-222
Smith
A-215
Turner
A-305
depositor
account_ branch_
number
name
A-101 Downtown
A-215 Mianus
A-102 Perryridge
A-305 Round Hill
A-201 Brighton
A-222 Redwood
A-217 Brighton
account
…
…
…
…
…
…
…
…
branch_name
Brighton
Downtown
Mianus
North Town
Perryridge
Pownal
Redwood
Round Hill
branch_city …
Brooklyn
Brooklyn
Horseneck
Rye
Horseneck
Bennington
PaloAlto
Horseneck
branch
…
…
…
…
…
…
…
…
{t | u ( depositor(u)  v (branch(v)  v[branch_city] = ‘Brooklyn’
w( account(w)  w[account_number] = u[account_number] 
w[branch_name] = v[branch_name]) )  t[customer_name] =
u[customer_name] ) }
Haichang Gao , Software School , Xidian University
73
DataBase System
x( P(x)  Q(x) )
Example Queries
 x(P(x)  Q(x) )
 Find all customers who have an account at all branches
located in Brooklyn city.
{t | u ( depositor(u)    v (branch(v)  v[branch_city] = ‘Brooklyn’ 
 w( account(w)  w[account_number] = u[account_number] 
w[branch_name] = v[branch_name]) )  t[customer_name] = u
[customer_name] ) }
等价转换
{t | u ( depositor(u)  v (branch(v)  v[branch_city] = ‘Brooklyn’
w( account(w)  w[account_number] = u[account_number] 
w[branch_name] = v[branch_name]) )  t[customer_name] =
u[customer_name] ) }
Haichang Gao , Software School , Xidian University
74
DataBase System
Example Queries
 Find the names of all customers having a loan from the
Perryridge branch, and the cities in which they live.
Banking DataBase:
branch (branch_name, branch_city, assets)
customer (customer_name, customer_street, customer_city)
depositor (customer_name, account_number)
account (account_number, branch_name, balance)
borrower (customer_name, loan_number)
loan (loan_number, branch_name, amount)
{t | u v w (customer(u)  borrower(v)  loan(w)  u[customer_name]
= v[customer_name]  v[loan_number] = w[loan_number] 
w[branch_name] = ‘Perryridge’  t[customer_name] =
u[customer_name]  t[customer_city] = u[customer_city] ) }
Haichang Gao , Software School , Xidian University
75
DataBase System
Safety of Expressions
 It
is possible to write tuple calculus expressions
that generate infinite(无穷的) relations.


For example, {t |  R(t) } results in an infinite relation if
the domain of any attribute of relation r is infinite
To guard against the problem, we restrict the set of
allowable expressions to safe expressions.

An expression {t | P(t)} in the tuple relational calculus is
safe if every component of t appears in one of the
relations, tuples, or constants that appear in P
 IN
Relational Algebra Operations we defined is
safety.
Haichang Gao , Software School , Xidian University
76
DataBase System
Homework 2
 5.1
 5.6
Haichang Gao , Software School , Xidian University
77