Algebra

Introduction to Database Systems

David Maier / Judy Cushing 1Relational Algebra, Calculus & SQL

Domains, Relations & Base RelVars (Ch. 5 Overview)Relational Algebra & Calculus,, & SQL (Ch. 6,7)Preview – Design Issues & Integrity Constraints

(Ch.8)A Design Review

• Ch. 5?– Each attribute implies a domain (data type, set of values)– Types do not imply physical representation (encap…)

• But at least one possible physical representation is suggested

– Specific data types are orthogonal to the relational model.– Scalar types have visible components (non-scalars don’t).– The relational model is strongly typed.– Some operators ( selector = < > := CAST AS )



Relations & RelVars• Relations (relation values)

A table is a concrete representation of a relation.– Tuple – a set of ordered pairs, no duplicates, unordered.– Relations have cardinality & degree.

• RelVars & Views define relations– The system catalog maintains the relvars.

• SQL is used to create – RelVars – CREATE DOMAIN, CREATE TABLE– Relations – INSERT, DELETE, UPDATE (tuples)



Type Checking

• J.CITY = P.CITY• JNAME | | PNAME• QTY * 100• QTY + 100• STATUS = 5• J.CITY < S.CITY• COLOR = P.CITY• J.CITY = P.CITY | | ‘burg’



Relational Algebra

Operates on relations, gives relations as results

Restrict (Select) -- subset of tuplesProject -- subset of columnsProduct -- all possible combinations of two tuples

AB

(Natural) Join: connecting tuples based on common value in same-named columns

XY

A XA YB XB Y

A1 B1A2 B1A3 B2

B1 C1B2 C2B3 C3

A1 B1 C1A2 B1 C1A3 B2 C2



Other Joins

• Theta Join -- join two relations but not on the equality operation

((S RENAME CITY AS SCITY) TIMES (P RENAME CITY AS PCITY) )

WHERE SCITY > PCITY

• Semi Join -- (natural) join two relations, but return only attributes of the first

((S SEMIJOIN (SP WHERE P# = # (‘P2’))



Standard Set Operations

Union, Intersection, Difference treat relations as sets of tuples

old_ext(inv# ext)

13 111813 111914 344316 3439

rings_at(inv# ext)

12 111813 111813 112014 340014 344315 344316 3439(inv# ext)

(inv# ext)

(inv#ext)



Division

Given 2 unary and 1 binary relation -- find tuples in the (first) unary relation

matched in the binary relation --w/ tuples in the other unary relation

ABC

A XA YA ZB XC Y

XZ

A



Division

phone_bk(person ext)

Randall 2116Ross 2116Ricard 2116Randall 2218Ross 2218Rivers 2218Ross 3972Ricard 3972Rivers 3972

r(ext)

2116

2118

= (person)

RandallRoss

s(person)

RossRicard

= (ext)

21163972

Analogya b is the largest q such that b * q a



Properties & Examples• Associative & Commutative --

UNION, INTERSECTION, TIMES, JOIN

• ((SP JOIN S) WHERE P# = P# (‘P2’)) {SNAME}• (((P WHERE COLOR = COLOR (‘Red’))

JOIN SP) {S#} JOIN S) {SNAME}• ((S {S#} DIVIDEBY P {P#} PER SP {S#, P#})

JOIN S {SNAME}• S{S#} DIVIDEBY (SP WHERE S# = S# (‘S2’)) {P#}

PER SP {S#,P#}• (((S RENAME S# AS SA) {SA, CITY} JOIN

(S RENAME S# AS SB) {SB, CITY} ) WHERE SA < SB) {SA,SB}



the Algebra enables writing relational expressions

• Retrieval• Update• Integrity constraints• Derived relvars (views)• Stability requirements (concurrency control)• Security constraints (scope of authorization)• Transformation rules -> OPTIMIZATION

((SP JOIN S) WHERE P# = P# (‘P2’)) {SNAME}((SP WHERE P# = P# (‘P2’)) JOIN S) {SNAME}



Miscellaneous

• Aggregates -- COUNT, SUM, AVG, MAX, MIN, ALL, ANY• SUMMARIZE SP PER SP {P#} ADD SUM (QTY) AS TOTQTY

SUMMARIZE (P JOIN SP) PER P {CITY} ADD COUNT AS NSP

• Relational Comparisons= equals/= not equals<= subset of (IN)< proper subset of>= superset> proper superset

• IS_EMPTY• GROUP, UNGROUP



Relational CalculusThe algebra - a language to describe

how to construct a new relation.The calculus - a language to write a

definition of that new relation. FORALL PX (PX.COLOR = COLOR (‘Red’))

EXISTS SPX (SX.S# = SX.S# AND SPX.P# = P# (‘P2’)

Procedural vs. Non-proceduralRelational Completeness



Query Exercises

• 6.13 Get full details of all projects.– J– JX– SELECT * FROM J

• 6.15 Get supplier numbers for suppliers who supply project J1.– ( SPJ WHERE J# = J# (‘J1’) ) {S#}– SPJX.S# WHERE SPJX.J# = J# (‘J1’) – SELECT DISTINCT SPJ.S#

FROM SPJ WHERE SPJ.J# = ‘J1’

• 6.18 Get supplier-number/part-number/project-number triples such that supplier part and project are all colocated).– (S JOIN P JOIN J) {S#, P#, J#}– (SX.S#, PX.P#, JX.J# ) WHERE SX.CITY = PX.CITY AND PX.CITY = JX.CITY AND

JX.CITY = SX.CITY

– SELECT S.S#, P.P#, J.J# FROM S,P,J WHERE S.CITY = P.CITY AND P.CITY = J.CITY.



Integrity Preview!

• An attribute value must match its type.• A primary key must be unique.• A tuple’s foreign key must be of the same type as its

“matching” primary key.• To create a record with a given foreign key, a record in the

corresponding table must have that value as its primary key.

• Many to many relationships cannot be modeled (directly). Find examples to support these in SPJ!



Study Questions

Which of the relational algebra, tuple calculus and domain calculus is SQL based on?

Can any relational query be expressed in a single SQL statement?

To what extent do these query languages go beyond the relational model?

Why might QueryByExample languages be easier to use than SQL for someone unfamiliar with the database scheme of a database?



Lab Assignment for Next Week

With your partner, finish creating and populating the SPJ database.

Run the assigned queries from Chapter 6.

SQL BNFhttp://cuiwww.unige.ch/~falquet/sdbd/langage/SQL92/BNFindex.html

Algebra

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