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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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