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Operators for a Relational Data ModelPowerPoint Presentation

Operators for a Relational Data Model

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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
- 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

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

- What is the projection here?
- Dimensional reduction (Z coordinate removed)

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

- 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 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

- What is the selection here?
- What do these foxy ladies want to wear today?

Notation

- σ< selection criteria > (R)
- New relation is thus a subset of the original relation

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 is the third operator
- Mathematical example:
- Equivalent terms
- Compact = Closed and Bounded

- Equivalent terms
- Renaming is used when combining relations
- Why would that be potentially necessary?
- Notation for this is ρattribute / attribute (R)

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

- What is this a Cartesian Product of?
- Truth values for P, Q, and R

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 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

- What 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?

Difference

- Difference is the sixth and final operator
- Mathematical example:
- Subsets

- Difference produces a subset not in common

Difference Example

- 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?

Friday

- Deriving relations and operators in our first order languages
- Combinations of relations
- Joins
- Algebras

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