Chapters 2 & 6

The Relational Model

2

The Relational Model A tabular data structure

Tables (relations) with unique names rows (tuples/entities/records) columns (attributes/fields)

Domain (or dom) of an attribute: a set of permitted atomic (non-decomposable) values

Tuples t r(R), where R = A1, A2, …, An, is an ordered set of values, i.e.,

t = < v1, v2, …, vn > where v1 dom(A1), v2 dom(A2), …, vn dom(An).

Relation r(R): a set of tuples r(R) dom(A1) dom(A2) … dom(An)

r is a relation instance

3

The Relational Model Database Schema

a relational DB schema consists of a set of relation schemas a relation schema has a (i) name and (ii) a finite set of attributes, where

each one is associated with a domain r(R) denotes a relation (instance) on the relation schema R

Definition. A candidate key K is a non-empty set of attributes which has the following properties:

a) uniqueness: K uniquely identifies a tuple in a relation, and

b) minimality: no proper subset of K satisfies the uniqueness constraint

• a primary key is a chosen candidate key

• a superkey satisfies (a) but may not satisfy (b)

• a relation may have more than one candidate key

• a foreign key in a relation schema is the primary key of another relation schema

Keys in the relational model K is a superkey for R if t1, t2 r(R), t1 t2, then t1[K] t2[K]

4

The Relational Model Relational Algebra

Operators used for manipulating relations, i.e., specifying retrieval/update requests (queries)

Relations in a relational DBMS are closed under the relational algebraic operations, since a

query result is itself is a relation

A procedural query language

5

The Relational Model Unary relational operators:

select () project () rename () assignment ()

Binary relational operators: cartesian produce () union () difference () intersection () natural join () division ()

Set Operations

6

The Relational Model Fundamental relational operators:

select () project () cartesian produce () union () difference () rename ()

Additional relational operators: intersection () natural join () division () assignment ()

7

The Relational Model Select ()

Selects the tuples (rows) from a relation r satisfying a certain selection condition C, i.e., c(r)

C is a Boolean expression on attributes of r

Resulting relation has the same attributes as r

e.g., (std_major = “CS” std_major = “Math”) std_gpa 3.5 (Student)

Project () Keeps only certain attributes specified in an attribute list L

from a relation r

Resultant relation has only those attributes of r specified in L

e.g., std_name, std_phone# (Student)

8

Relational Algebra

A = “a¹” (r¹) A B Ca1 b1 c1

a1 b3 c1

A,C (r¹)A Ca1 c1

a2 c2

r¹A B Ca1 b1 c1

a2 b2 c2

a1 b3 c1

Example.

9

The Relational Model Cartesian Product ()

combines information from two relations

r¹(A¹, A2, …, An) r2 (B¹, B2, …, Bm) = r3(A¹, …, An, B¹, …, Bm), and t r3, t[A¹, …, An] r1, t[B¹, …, Bm] r2

if |r¹| = i and |r2| = j

Union ()

r1 r2: combines tuples in r1 with those tuples in r2

duplicate tuples are eliminated

the two involved relations must be union-compatible

union-compatible:

1) same arity (degree), i.e., same number of attributes

2) attributes in the corresponding columns must have the same domain

, then |r1 r2| = i j

10

Relational AlgebraExample (continued).

r¹

A B C

a1 b1 c1

a2 b2 c2

a1 b3 c1

r2

A B C

a1 b3 c1

a2 b1 c1

r¹ r2

r¹.A r2.Ar¹.B r¹.C r2.B r2.C

a¹ a¹b¹ c¹ b3 c1

a¹ a2b¹ c¹ b1 c1

a2 a¹b2 c2 b3 c1

a2 a2b2 c2 b1 c1

a¹ a¹b3 c¹ b3 c1

a¹ a2b3 c¹ b1 c1

r¹ r2

A B C

a1 b1 c1

a2 b2 c2

a1 b3 c1

a2 b1 c1

11

The Relational Model

Set Difference ()

r1 r2 : contains tuples which occur in r1 but not in r2

the two involved relations must be union-compatible

Rename ()

s(r) returns relation r under the name s, or

BA(r) returns attribute A under the attribute name B

12

Relational Algebra

Example (continued).

r¹

A B C

a1 b1 c1

a2 b2 c2

a1 b3 c1

r2

A B C

a1 b3 c1

a2 b1 c1

s(r¹)

A B C

a1 b1 c1

a2 b2 c2

a1 b3 c1

r¹ r2

A B C

a1 b1 c1

a2 b2 c2

=

13

The Relational Model Additional Relational Operations

simplify queries constructed by using the fundamental operators

more convenient since less operators involved

do not add expressive power to the relational model

Additional Operations: Set Intersection ()

• r1 r2 contains common tuples in r1 and r2

• the two involved relations must be union-compatible

• can be rewritten using set difference, i.e.,

r s = r (r s)

14

Relational Algebra

Example (continued).

r¹

A B C

a1 b1 c1

a2 b2 c2

a1 b3 c1

r2

A B C

a1 b3 c1

a2 b1 c1

r1 r2

A B C

a1 b3 c1

15

The Relational Model – Additional Operators

Natural Join (): a combination of , , in that order

: merge two relations into one

: select equal values on common attributes

: remove duplicated columns

A “” becomes “” if the two operand relations have no common attributes, e.g.,

r(A, B) s(C, D) = r(A, B) s(C, D)

Rewrite using , and :

r(R) s(S) = R S (r.A1 = s.A1 ... r.An = s.An (r s))

where R S = {A¹, …, An}

16

Relational Algebra

r¹

A B Ca1 b1 c1

a2 b2 c2

a1 b3 c1

r3

B D

b3 d4

b1 d5

b1 d3

Example (continued).

A B Ca1 b1 c1

Dd3

a1 b1 c1 d5

a1 b3 c1 d4

r1 r3

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Assignment ()

assigns a relation with a relation variable, a convenient way to express complex queries, e.g.,

V R S(r)

Division ()

retrieve each (resultant) tuple in the dividend relation that “contain” all the tuple values of the divisor relation

attribute(divisor) attribute(dividend)

resultant schema has attribute(dividend) – attribute(divisor),

e.g., Has_taken(Name, Course#, Status) Core_course(Course#)

The Relational Model – Additional Operators

18

Relational Algebra

StatusName Has_taken Core_course

Has_taken

Steve

Name Course#Math 112

StatusFreshman

DanSteve CS 103

CS 142Freshman

Junior

DanDan Math 112

CS 143JuniorJunior

Linda CS 142 Freshman

Core_courseCourse#

MATH 112CS 142

Example.

JuniorDan

19

Division Operation

Example. Let r and s be the following relations:

r s

A

B

1

2

A B

12311134612

r s

20

Division Operation

r s

A B CA B

aaaaaaaa

C D

aabababb

E

11113111

D

ab

E

11

r s

Example. Let r and s be the following relations:

a a

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Division Operation

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)

r s = { t | t R-S(r) u s ( tu r ) }

r s

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Division Operation Property

Let q = 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 = R-S (r) – R-S ((R-S (r) × s) – R-S,S(r))

To see why

R-S,S(r) simply reorders attributes of r

R-S(R-S (r) × s) – R-S,S(r)), i.e., (iii), gives those tuples t in

R-S (r) such that for some tuple u s, tu r.

(i)

(ii)

(iii)

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Sample Query: “Find all customers who have an account at all branches located in Brooklyn city.”

branch (branch-name, branch-city, assets)

depositor (customer-name, account-number)

account (account-number, branch-name, balance)

Division Operation

(branch-city = “Brooklyn” (branch))branch-name

(depositor account)customer-name, branch-name

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Outer Join An extension of the join operation that avoids loss of

information.

Computes the (natural) join and then adds tuples from one relation that does not match tuples in the

other relation to the result of the join.

Uses null values:

null signifies that the value is unknown or does not exist

All comparisons involving null are (roughly speaking) false by definition.

• Will study precise meaning of comparisons with nulls later

25

Outer Join

Natural Join: loan Borrower

Left Outer Join: loan Borrower

loan-number amount

L-170L-230

30004000

customer-name

JonesSmith

branch-name

DowntownRedwood

loan-number amount

L-170L-230L-260

300040001700

customer-name

JonesSmithnull

branch-name

DowntownRedwoodPerryridge

loanloan-number amount

L-170L-230L-260

300040001700

borrower

customer-name loan-number

JonesSmithHayes

L-170L-230L-155

branch-name

DowntownRedwoodPerryridge

Example

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Outer Join

Right Outer Join: loan borrower

Full Outer Join: loan borrower

loan-number amount

L-170L-230L-155

30004000null

customer-name

JonesSmithHayes

branch-name

DowntownRedwoodnull

loan-number amount

L-170L-230L-260L-155

300040001700null

customer-name

JonesSmithnullHayes

branch-name

DowntownRedwoodPerryridgenull

loanloan-number amount

L-170L-230L-260

300040001700

borrower

customer-name loan-number

JonesSmithHayes

L-170L-230L-155

branch-name

DowntownRedwoodPerryridge

Example

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Left outer join ( ) defined by other algebraic operators

r s = (r s)

where (null, …, null) is on the schema S - R

Right outer join ( ) defined by other algebraic operators

r s = (r s) ( (s - S(s r)) { (null, …, null) } )

where (null, …, null) is on the schema R - S

Full outer join ( ) defined by left and right outer joins

r s = (r s) (r s)

Outer Join

( (r - R(r s)) { (null, …, null) } )

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Null Values It is possible for tuples to have a null value, denoted by null, for

some of their attributes.

Null signifies an unknown value or that a value does not exist

The result of any arithmetic expression involving null is null.

For duplicate elimination and grouping, null is treated like avalue, and two nulls are assumed to be the same.

Alternative: assume each null is different from each other (adopted)

Both are arbitrary decisions

Relational (comparison) operations (such as >, <, =, etc.) involving null values return the special truth value unknown, which is null.

29

Null Values Three-valued logic using the truth value unknown:

AND: (true and unknown) = unknown

(false and unknown)= false

(unknown and unknown)= unknown

OR: (unknown or true) = true (unknown or false) = unknown (unknown or unknown) = unknown

NOT: (not unknown) = unknown

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Null Values Selection: Comparisons with null values return the special

truth value unknown, i.e., not selected

Natural Join: joining tuples on null values cause mismatch

Projection: treating nulls like any other values when eliminating duplicates; our assumption is: difference

Union, Intersect, Difference: treating nulls just as the projection does, i.e., null values in corresponding columns in two tuples are treated as different

Outer join: behaving just like join operations, except on tuples that do not occur in the join results, which may be added

to the result, depending on the operation.

31

Deletion

A deletedelete request is expressed similarly to a query, except instead of displaying tuples to the user, the selected tuples are removed from the database.

Can delete only whole tuples; cannot delete values on only particular attributes

A deletion is expressed in relational algebra by:

r r – E

where r is a relation and E is a relational algebra query.

32

Deletion

Delete all account records in the Perryridge branch.

account account – branch-name = “Perryridge” (account)

Delete all loan records with amount in the range of 0 to 50.

loan loan – amount 0and amount 50 (loan)

Delete all accounts at branches located in Needham.

r1 branch-city = “Needham” (account branch)

r2 account-number, branch-name, balance (r1)

account account – r2

r3 customer-name, account-number (r2 depositor)

depositor depositor – r3

Example. branch (branch-name, branch-city, assets) account (account-number, branch-name, balance) loan (loan-number, branch-name, amount) depositor (customer-name, account-number)

33

Insertion To insert data into a relation, we either:

specify a tuple to be inserted, or

write a query whose result is a set of tuples to be inserted

In relational algebra, an insertion is expressed by:

r r E

where r is a relation and E is a relational algebra expression.

The insertion of a single tuple is expressed by letting E be a constant relation containing one tuple.

34

Insertion

Insert information in the database specifying that “Smith” has $1200 in account A-973 at the “Perryridge” branch.

account account { (“A-973”, “Perryridge”, 1200) }

depositor depositor { (“Smith”, “A-973”) }

Provide as a gift for all loan customers in the “Perryridge” branch, a $200 savings account. Let the loan number serve as the

account number for the new savings account.

r1 (branch-name = “Perryridge” (borrower loan))

account account loan-number, branch-name, 200 (r1)

depositor depositor customer-name, loan-number(r1)

Example. account (account-number, branch-name, balance) depositor (customer-name, account-number)

borrower (customer-name, loan-number) loan (loan-number, branch-name, amount)

35

Modification A mechanism to change a value in a tuple without changing

all the values in the tuple.

Use the generalized projection operator to do this task.

r F1, F2, …, Fn, (r)

Each Fi (1 i n) is either the ith attribute of r, if the ith attribute is not updated, or a predicate involving the ith

attribute, if it is the attribute to be updated.

Fi is an expression, involving only constants and the attributes of r, which gives the new value for the attribute.

36

Modification

Make interest payments by increasing all balances by 5%

account account-number, branch-name, balance * 1.05 (account)

Pay all accounts with balances over $10,000 6% interest and pay all others 5%

account account-number, branch-name, balance*1.06 (balance 10000 (account)) account-number, branch-name, balance*1.05 (balance 10000 (account))

Example. account (account-number, branch-name, balance)

37

The Relational Model Database Updates

A DB application provides a mechanism for updating data Method for updating data: use an interactive query/update language Updating means inserting, deleting, and modifying db data

Delete• Removes selected tuples from db tables

• Relational algebra expression: r r E, where r is a relation & E is a relational algebra query

Insertion• Add tuples into db tables

t = <a1, a2,…, an>, a tuple to be inserted into relation r(A1,…, An), ai dom(Ai), 1 i n

• Relational algebra expression: r r E

Modification ()• Changes a value in a tuple

• Relational algebra expression: A E(r), where A is an attr of relation r, & E is an arithmetic expression which includes constants & attrs of r.

38

The Relational Model Tuple Relational Calculus

A nonprocedural (declarative) query language

Formal DB language based on 1st order predicate calculus

Tuple variables range over tuples of a relation

Attribute values: t[A], value of attribute A in tuple variable t

Tuple calculus expressions:

{ t | P(t) }

i.e., t: predicate (formula) P is true for t

• Quantifier: (universal/for all), (existential/there exists)

• Free/bound tuple variable: a variable not quantified/quantified by a quantifier

e.g., t r1 s r2

• Formulas are made up of atoms

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The Relational Model Tuple Relational Calculus

Atoms are of the following types:

• t r, where t is a tuple variable and r is a relation

• t1[X] t2[Y], where

– t1 and t2 are (same) tuple variables

– X, an attribute of t1 and Y, an attribute of t2

– { <, , , , >, }

– dom(X) and dom(Y) are -compatible

• t[X] C, where

– t is a tuple variable

– X, an attribute of t

– C, a constant in dom(X)

– , same as above, a comparison operator

40

The Relational Model Tuple Calculus Expressions

Well-formed formula (wff):

1. an atom is wff

2. if P is a wff, then

P is a wff

(P) is a wff

3. if P1 and P2 are wffs, then

P1 P2 is a wff

P1 P2 is a wff

P1 P2 is a wff

4. if P(t) is a wff, where t is free, then

t r (P(t)) is a wff

t r (P(t)) is a wff

41

The Relational Model Relational Algebra Expressions vs. Tuple Calculus Expressions

Relational Alg. Expression Tuple Calculus Expression

Select: C(r) { t | t r C’ }, where C’ is the condition C with each appearance of attribute A of r

replaced by t[A]

Project: A1, …, An(r) { t | u r (t[A1] = u[A1] ... t[An] = u[An])}

Cartesian: r(R) s(S) { t | x r (y s

(t[A1] = x[A1] ... t[An] = x[An] t[B1] = y[B1] … t[Bm] = y[Bm])) }

where R = {A1, …, An} and S = {B1, …, Bm}

Union: r s { t | t r t s }

Difference: r s { t | t r (t s) }

Product

42

Sample Tuple Calculus Queries

Find the loan-number, branch-name, and amount for loans of over $1200.

{ t | t loan t [amount] 1200 }

Find the loan number for each loan of an amount greater than $1200.

{ t | n loan (t[loan-number] = n[loan-number] n[amount] 1200) }

Example. loan (loan-number, branch-name, amount)

43

Find the names of all customers having a loan, an account, or both at the bank.

{ t | b borrower (t [customer-name] = b[customer-name]) d depositor (t [customer-name] = d[customer-name])}

Find the names of all customers who have a loan and an account at the bank.

{ t | b borrower (t [customer-name] = b[customer-name]) d depositor (t [customer-name] = d[customer-name]) }

Example. borrower (customer-name, loan-number) depositor (customer-name, account-number)

Sample Tuple Calculus Queries

44

Find the names of all customers having a loan at the Perryridge branch.

{ t | b borrower (t [customer-name] = b[customer-name] n loan (b[loan-number] = n[loan-number]

n[branch-name] = “Perryridge”)) }

Find the names of all customers who have a loan at the Perryridge branch, but no account at any branch of the bank.

{ t | b borrower (t [customer-name] = b[customer-name] n loan (b[loan-number] = n[loan-number]

n[branch-name] = “Perryridge”)) d depositor (d[customer-name] = t [customer-name])}

Example. borrower (customer-name, loan-number) loan (loan-number, branch-name, amount) depositor (customer-name, account-number)

/

Sample Tuple Calculus Queries

45

Find the names of all customers having a loan from the Perryridge branch, and the cities they live in.

{ t | n loan (n[branch-name] = “Perryridge” b borrower (b[loan-number] = n[loan-number] t [customer-name] = b[customer-name] c customer (c[customer-name] = b[customer-name]

t [customer-city] = c[customer-city]))) }

Example.

loan (loan-number, branch-name, amount) borrower (customer-name, loan-number)

customer (customer-name, customer-street, customer-city)

Sample Tuple Calculus Queries

46

Find the names and cities of all customers who have an account at all branches located in Brooklyn.

{ t | c customer (t [customer-name] = c[customer-name] t [customer-city] = c[customer-city] b branch (b[branch-city] = “Brooklyn” a account (a[branch-name] = b[branch-name] d depositor (d[account-number] = a[account-number] d[customer-name] = c[customer-name]))))}

Example.

customer (customer-name, customer-street, customer-city) depositor (customer-name, account-number) account (account-number, branch-name, balance)

branch (branch-name, branch-city, assets)

Sample Tuple Calculus Queries

47

The Relational Model Safe Tuple Calculus Expressions

Expressions generate infinite relations, e.g., { t | (t r) }, is not safe

• An expression E: { t | P(t) } is safe if each value in E dom(P)

• dom(P) = { Constants in P } { Attribute values of relations in P }