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Schedule• Today:

Relational Algebra. Read Chapter 5 to page 199.

• Next SQL Queries. Read Sections 6.1-6.2.

• And then Subqueries, Grouping and Aggregation. Read Sections 6.3-6.4.

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Relational Algebra

• The relational algebra is a precise mathematical notation and set of rules for manipulating “relations”.

• SQL is basically a more human readable form of the relational algebra.

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“Core” Relational AlgebraA small set of operators that allow us to manipulate

relations in limited but useful ways. The operators are:

1. Union, intersection, and difference: the usual set operators. But the relation schemas must be the same.

2. Selection: Picking certain rows from a relation.

3. Projection: Picking certain columns.

4. Products and joins: Composing relations in useful ways.

5. Renaming of relations and their attributes.

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Relational Algebra

SELECT

π PROJECT

X CARTESIAN PRODUCT

NATURAL JOIN

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SelectionR1 = C(R2)

where C is a condition involving the attributes of relation R2.

ExampleRelation Sells:

JoeMenu = bar=Joe's(Sells)

bar beer priceJoe's Bud 2.50Joe's Miller 2.75Sue's Bud 2.50Sue's Coors 3.00

bar beer priceJoe's Bud 2.50Joe's Miller 2.75

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Product

R = R1 R2

pairs each tuple t1 of R1 with each tuple t2 of R2 and puts in R a tuple t1t2.

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Natural-Join

R = R1 R2

is equivalent to R = C(R1 R2) where c is the condition that the the values of the attributes that R1 and R2 have in common must match.

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ExampleSells = Bars =

BarInfo = Sells Bars

bar beer priceJoe's Bud 2.50Joe's Miller 2.75Sue's Bud 2.50Sue's Coors 3.00

bar addrJoe's Maple St.Sue's River Rd.

bar beer price bar addrJoe's Bud 2.50 Joe's Maple St.Joe's Miller 2.75 Joe's Maple St.Sue's Bud 2.50 Sue's River Rd.Sue's Coors 3.00 Sue's River Rd.

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Combining Operations

Algebra =1. Universe or domain of objects2. Operators for constructing expressions.For relational algebra:1. Domain = variables standing for relations

+ finite, constant relations.2. Expressions constructed by applying one

of the operators + parentheses.• Query = expression of relational algebra.

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Bag Semantics

A relation (in SQL, at least) is really a bag or multiset.

• It may contain the same tuple more than once, although there is no specified order (unlike a list).

• Example: {1,2,1,3} is a bag and not a set.• Select, project, and join work for bags as

well as sets. Just work on a tuple-by-tuple basis, and don't

eliminate duplicates.

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Bag Union

Sum the times an element appears in the two bags.• Example: {1,2,1} {1,2,3,3} = {1,1,1,2,2,3,3}.

Bag IntersectionTake the minimum of the number of occurrences in each

bag.• Example: {1,2,1} {1,2,3,3} = {1,2}.

Bag DifferenceProper-subtract the number of occurrences in the two bags.• Example: {1,2,1} – {1,2,3,3} = {1}.

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Duplicate Elimination(R) = relation with one copy of each tuple that appears one

or more times in R.

ExampleR =

A B1 23 41 2

(R) =A B1 23 4

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Bank Database Schema

• Branch = (branch-name, branch-city, assets)• Customer = (customer-name, customer-street,

customer-city)• Account = (branch-name, account#, balance)• Depositor = (customer-name, account#)• Loan = (branch-name, loan#, amount)• Borrower = (customer-name, loan#)

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The Customer Table

Customer-name C-Street C-city

Bob 123 Third St San Jose

Carol 456 Main St Santa Clara

Ted 89 Blossom Ave Los Gatos

Alice 64 Longwalk Dr Oakland

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The Account Table

Branch-name Account# Balance

Oakland 101 2000

SJ-Main 205 7500

SJ-Main 207 4500

Santa Clara 311 3100

SJ-West 251 850

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The Loan Table

Branch-name loan# Amount

Oakland 2301 5000

SJ-Main 5155 700

SJ-Main 5709 9000

Santa Clara 1541 1800

SJ-West 4321 250

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The Depositor Table

Customer-name Account#

Bob 207

Carol 311

Ted 205

Alice 101

Bob 251

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Queries on the Loan Table

• Loan = (branch-name, loan#, amount)

• Find the names of all the branches in the Loan relation

select branch-namefrom Loan

(branch-name) (Loan)

Branch-name

Oakland

SJ-Main

SJ-Main

Santa Clara

SJ-West

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More Queries on the Loan Table

select *from Loanwhere amount > 3000

 (amount>3000) (Loan)

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• Find the loan numbers for all loans made at the Oakland branch with loan amounts greater than 1200. 

select loan#from Loanwhere branch-name="Oakland"

and amount>1200(loan#) (branch-name=“Oakland”^amount>1200) (Loan)

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Cross Product

Select A1, A2 from R1, R2Result R = R1 R2

• pairs each tuple t1 of R1 with each tuple t2 of R2 and puts in R a tuple t1t2.

(A1, A2) (R1 R2)

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Natural Join• Find the name of customers with an account at

the Oakland branch.select customer-namefrom Depositor, Accountwhere Depositor.account# = Account.account# and branch-name = "Oakland"

(cust-name) (b-name=“Oakland”) (Depositor Account)