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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

Operators for a Relational Data Model

Matt Dube

Doctoral Student, Spatial Information Science and Engineering


Monday s class
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
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
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


Projection
Projection

  • 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


Projection example
Projection Example

  • What is the projection here?

  • Dimensional reduction (Z coordinate removed)


Notation
Notation

  • π< pertinent attributes > (R)

  • New relation is thus a subset of a different space

  • That different space is a component of the original domain



Problem with projection
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)


Selection
Selection

  • 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


Selection example
Selection Example

  • What is the selection here?

  • What do these foxy ladies want to wear today?


Notation1
Notation

  • σ< selection criteria > (R)

  • New relation is thus a subset of the original relation



Properties of selection
Properties of Selection

  • 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
Renaming

  • 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)


Cartesian product
Cartesian Product

  • 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


Cartesian product example
Cartesian Product Example

  • What is this a Cartesian Product of?

  • Truth values for P, Q, and R



Properties of a cartesian product
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?


Union
Union

  • Union is the fifth operator

  • Mathematical Example:

    • Addition of positive integers is a natural union

    • Addition of sets

  • Union thus takes relations and binds them together


Union example
Union Example

  • What is this a union of?

  • The 50 States and Puerto Rico and the Virgin Islands


What does a union have
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?



Difference
Difference

  • Difference is the sixth and final operator

  • Mathematical example:

    • Subsets

  • Difference produces a subset not in common


Difference example
Difference Example

  • What is the difference here?


Unary and n ary operators
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?


Friday
Friday

  • Deriving relations and operators in our first order languages

  • Combinations of relations

  • Joins

  • Algebras


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