Operators for a Relational Data Model

Matt DubeDoctoral Student, Spatial Information Science and Engineering

Monday’s Class Mathematical definitions that underlie the relational

data model: Domain: set of inputs for a particular attribute Type: structure of the members of that set Cartesian Product: combination of members of one

set with every member of another set…and another…and another

Relation: subset of the Cartesian product of the attribute domains

Key: unique identifying attributes for a relation

Discussion of Assignment

Think of an example in your particular discipline where a relational data model might be helpful.

What are the attributes?

What are their domains?

What would you key the database with?

Why?

Stepping Up from Relations

Having only useful data was a motivating concern for us

How do we go about that?Operators are the critical component that

allows us to transform a relational database of large size to one which is more manageable

There are six of these operators to be concerned with

ProjectionProjection is the first of the operatorsMathematical example:

Projection is the “shadow” of a vector of any sort onto a lower dimensional surface.

Think of a right triangle: the lower leg is always shorter than the hypotenuse (why?)

Projection thus represents only considering certain attributes of interest

Projection ExampleWhat is the projection here?

Dimensional reduction (Z coordinate removed)

Notationπ< pertinent attributes > (R)

New relation is thus a subset of a different space

That different space is a component of the original domain

Table ProjectionName Major Schoo

lGende

rAndrew ECE UMain

eM

Guillaume

GIS UMaine

M

Valeryia ECO UMaine

F

Lisa GIS UMaine

F

Eric BUA UMaine

M

Chris SIE UMaine

M

Name GenderAndrew M

Guillaume MValeryia F

Lisa FEric M

Chris M

Problem with Projection What dictates the usefulness of a projection?

Is the key involved?

What if a key isn’t involved?

If the key isn’t involved, duplicates are removed to preserve relation status = missing data!

Order not important…attributes can be listed in any order in the projection function (analogous to rotation)

SelectionSelection is the second operator, and is the

converse of projectionMathematical example:

Intersections in a Venn DiagramSelection takes a list of specific properties and

find things which satisfy that list

Selection ExampleWhat is the selection here?

What do these foxy ladies want to wear today?

Notationσ< selection criteria > (R)

New relation is thus a subset of the original relation

Table SelectionName Major Schoo

lGende

rAndrew ECE UMain

eM

Guillaume

GIS UMaine

M

Valeryia ECO UMaine

F

Lisa GIS UMaine

F

Eric BUA UMaine

M

Chris SIE UMaine

M

Name Major

School

Gender

Andrew ECE UMaine

M

Valeryia ECO UMaine

F

Eric BUA UMaine

M

Chris SIE UMaine

M

Properties of SelectionSelection is in the same application space as

the original relationKey structure is thus the sameSelection is associative

Associativity: Being able to interchange the groupings Addition and multiplication are associative

operators you are familiar with already

RenamingRenaming is the third operatorMathematical example:

Equivalent termsCompact = Closed and Bounded

Renaming is used when combining relationsWhy would that be potentially necessary?

Notation for this is ρ attribute / attribute (R)

Cartesian ProductWe went over this a bit mathematically, but

now we are going to apply it to relations themselves

Mathematical Example: The X,Y Plane (or the X,Y,Z space, or any other

similar type of space)Take all possible combinations of relation

records between 2 or more relations

Cartesian Product Example

What is this a Cartesian Product of?

Truth values for P, Q, and R

P Q RT T TT T FT F TT F FF T TF T FF F TF F F

PTF

PTF

QTF

RTF

Cartesian ProductName PetGeorge Fluffy

Karl Rover

State CityDelaware DoverOregon Corvallis

Name Pet State CityGeorge Fluffy Delawar

eDover

Karl Rover Delaware

Dover

George Fluffy Oregon CorvallisKarl Rover Oregon Corvallis

Properties of a Cartesian Product

How big will a Cartesian Product be? Treat this generally:

Relation R has x rows and y columns Relation S has z rows and w columns

R x S has x * z rows and y + w columns Why? The key of a Cartesian product needs to involve at least

one attribute from both R and S. Why?

UnionUnion is the fifth operatorMathematical Example:

Addition of positive integers is a natural union Addition of sets

Union thus takes relations and binds them together

Union ExampleWhat is this a union of?

The 50 States and Puerto Rico and the Virgin Islands

What does a Union have?

Unioned attributes always have identical domains Why?

Do unioned attributes have to have identical names? No Think of unioning two sets together. Did those sets have the

same names? Unions are commutative

Commutativity: Changing the order is irrelevant Addition and multiplication are commutative operators Difference between this and associativity?

Commutative vs. Associative

DifferenceDifference is the sixth and final operatorMathematical example:

Subsets

Difference produces a subset not in common

Difference ExampleWhat is the difference here?

Unary and N-ary Operators

Unary operators only have one operand (in this case they only involve one relation)

Projection, Selection, RenamingN-ary operators involve N operands (in this

case they involve N relations)Cartesian Product, Union, DifferenceCan you classify the six?

FridayDeriving relations and operators in our first

order languagesCombinations of relations JoinsAlgebras