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# Operators for a Relational Data Model - PowerPoint PPT Presentation

Operators for a Relational Data Model. Matt Dube Doctoral 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

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### Operators for a Relational Data Model

Matt Dube

Doctoral Student, Spatial Information Science and Engineering

• 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

• 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?

• 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

• Projection is the first of the operators

• Mathematical 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

• What is the projection here?

• Dimensional reduction (Z coordinate removed)

• π< pertinent attributes > (R)

• New relation is thus a subset of a different space

• That different space is a component of the original domain

• 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)

• Selection is the second operator, and is the converse of projection

• Mathematical example:

• Intersections in a Venn Diagram

• Selection takes a list of specific properties and find things which satisfy that list

• What is the selection here?

• What do these foxy ladies want to wear today?

• σ< selection criteria > (R)

• New relation is thus a subset of the original relation

• Selection is in the same application space as the original relation

• Key structure is thus the same

• Selection is associative

• Associativity:

• Being able to interchange the groupings

• Addition and multiplication are associative operators you are familiar with already

• Renaming is the third operator

• Mathematical example:

• Equivalent terms

• Compact = Closed and Bounded

• Renaming is used when combining relations

• Why would that be potentially necessary?

• Notation for this is ρattribute / attribute (R)

• We 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

• What is this a Cartesian Product of?

• Truth values for P, Q, and R

• 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?

• Union is the fifth operator

• Mathematical Example:

• Addition of positive integers is a natural union

• Union thus takes relations and binds them together

• What is this a union of?

• The 50 States and Puerto Rico and the Virgin Islands

• 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?

• Difference is the sixth and final operator

• Mathematical example:

• Subsets

• Difference produces a subset not in common

• What 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, Renaming

• N-ary operators involve N operands (in this case they involve N relations)

• Cartesian Product, Union, Difference

• Can you classify the six?

• Deriving relations and operators in our first order languages

• Combinations of relations

• Joins

• Algebras