Relational Math
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Relational Algebra
 The rules for combining one or more numbers
(or symbols standing in for numbers) to obtain
another number is called algebra.
 For example, the rules for combining one or
more Boolean variables (expressions which are
either true or false) to obtain another Boolean is
called Boolean algebra.
 The rules for combining one or more relations to
obtain another relation is called relational
algebra.
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Operations
 The specific ways of combining
elements are known as operations.
 E.g. addition is an algebraic operation
 E.g. ANDing is a Boolean operation
unary if they act on
one element and binary if they act on two
 Operations are called
elements.
 E.g square root is a unary algebraic operation.
 E.g. addition is a binary algebraic operation.
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Relation  Table

Relational algebra sounds so abstract.
 Recall a representation of a relation is a table.
 So relational algebra means that we do stuff to
tables and get other tables out.
 A relational database is made of tables.
 Relational algebra tells us how to operate on tables.
 That is, relational algebra tells us what a Data
Manipulation Language (DML) should do.
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Synonym
 Recall our old synonyms
 Table  Relation  File
 Row  Tuple  Record
 Column  Attribute  Property  Field
 A new synonym pair is
 Condition  Predicate
 A condition is a Boolean, an expression that is true
or false, e.g. Salary > 600000 or Name=“Smith”
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Selection/Restriction
 A selection (a.k.a. restriction) picks out those
rows from a table that meet some condition.
 Example: Let us select from the Customer table
those people who are from PA.
 predicate ( R )
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Selection Example: Customers from PA
(Design)
Customer.* refers to
all of the columns.
The condition (predicate)
selecting out particular rows.
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Selection Example: PA Only (DataSheet)
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Selection Example: PA Only (SQL)
Condition
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Condition can be compound.
 The selection condition may be a compound
condition.
 ConditionA AND ConditionB
 ConditionA OR ConditionB
 Example: Let us select from the Customer table
those people who from Philadelphia and from PA.
(There are other Philadelphias, e.g. in Mississippi)
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Added Some New Customers
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Selection Example: Philadelphia AND PA
(Design)
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ANDed conditions are
entered on the same line.
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Selection Example: Philadelphia AND PA
(DataSheet)
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Selection Example: Philadelphia AND PA
(SQL)
ANDed conditions
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Selection Example: Customers from PA or
NJ (Design)
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ORed conditions are
entered on separate
lines.
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Selection Example: Customers from PA or
NJ (DataSheet)
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Selection Example: Customers from PA or
NJ (SQL)
ORed Conditions
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Projection
 The projection operator picks out a set of
columns that will belong to the resulting table.
 Recall the concept of views in which certain fields
would be hidden from certain users.
 Example: Let us project from the Customer table
the first and last names.
 column1,column2,… ( R )
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Projection Example: Customer’s first and
last names (Design)
Choose columns
and check to
show them.
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Projection Example: Customer’s first and
last names (DataSheet)
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Projection Example: Customer’s first and
last names (SQL)
Columns that will appear.
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Union Compatible
 Think of the records in a table as elements of
a set.
 If two sets have the same sorts of records,
that is, the same fields in the same order or
minimally the same type fields in the same
order, then the sets are said to be unioncompatible.
 Then you can consider forming
 The union of the two sets
 The intersection of the two sets
 The set difference
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A Simpsons Database
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Union
 The union of set A and set B contains all of the
elements of set A as well as all of the elements of set
B
 If an element belongs to set A and set B, the union
contains only one copy of it.
 Example: let us make a table containing all of the
names from the Character and RealPerson tables
 FirstName,LastName(Character)  FirstName,LastName(RealPerson)
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Union Example: Character and RealPerson
names
Step 1 would be to create union-compatible tables using
projection. Step 2 would be to take the union of these
tables.
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Union Example: Character and RealPerson
names (DataSheet)
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Union Example: Character and RealPerson
names (DataSheet)
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Union Example: Character and RealPerson
names (No Design )
 Query-By-Example (QEB) which is what we do in
Design View takes the join as its principle binary
operation.
 While the union is a more fundamental binary
operation in Relational algebra, the join is the
more common operation in querying.
 SQL does have the union operation!
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Union Example: Character and RealPerson
names
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UNION ALL
 UNION ALL is a variation on the UNION
operation that does not eliminate duplicate
records from the results.
 It is somewhat faster because the system does not
have to look for the possibility of duplications.
 The result is “weird” in that we usually do not want
duplicate records.
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Intersection
 The intersection of set A and set B contains only the
elements that belong both to set A and to set B.
 Example: let us make a table containing all of the
names of people who play themselves on the
Simpsons.
 
FirstName,LastName(Character)

FirstName,LastName(RealPerson)
 Again the first step is to make “union-compatible”
tables.
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Intersection Example: People playing
themselves (DataSheet)
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Intersection Example: People playing
themselves (Design, step 1)
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Intersection Example: People playing
themselves (Design, step 2)
Dragging a field icon
from one table to
another establishes a
relationship. Right
click on a line to
remove a relationship
from the query.
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Intersection Example: People playing
themselves (SQL)
SQL has an INTERSECT operation like its UNION
operation, but it is not supported by Access.
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Intersection Example: People playing
themselves (Design, version 2)
concatenation
subquery
Uses concatenation and a subquery.
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Intersection Example: People playing
themselves (SQL, version 2)
Note: Access adds lots of parentheses.
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Set Difference
 The set difference of Set A and Set B is all of the
elements in Set A that are not also elements of set
B.
 Example: Simpsons characters who are not real
people.
 FirstName,LastName(Character) -FirstName,LastName(RealPerson)
 Again the first step is to make “union-compatible”
tables.
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Set Difference Example: Characters that
are not real people (DataSheet)
Ernest Borgnine and
James Brown removed.
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Set Difference Example: Characters that
are not real people (Design)
Same as the second version of the
Intersection query except IN  NOT IN
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Set Difference Example: Characters that
are not real people (SQL)
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Cartesian Product
 A row in the Cartesian product of
Table A and Table B is the concatenation of a
row from Table A and a row from Table B.
 All possible combinations of a row from A and
a row from B are made.
AB
 On its own the Cartesian product is not very
useful, but it is the first ingredient in a join,
which is very useful in querying relational
databases.
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How big is the Cartesian product?
Degree(B)
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Cardinality(B)
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Cardinality(AB) = Cardinality(A) * Cardinality(B)
Degree(AB) = Degree(A) + Degree(B)
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Perform a Selection on the Cartesian
Product
 Recall that
 (Most of) our tables correspond to entities.
 Entities have relationships.
 Relationships are realized by having fields in two tables
take values from the same domain.
 E.g. That a Character is voiced by a Real Person is represented
by having the PersonID (which identifies a person in the
RealPerson Table) appear the Character Table.
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Perform a Selection on the Cartesian
Product (Cont.)
 The Cartesian product of the Character and
RealPerson Tables has rows in which the
person voices the character and rows in which
the person does not voice the character.
 What distinguishes the former is that the
Character.PersonID matches the
RealPerson.PersonID.
 We can use this condition (predicate) to select
out the meaningful rows.
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After the selection
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Now we have a table with two identical columns. We
can eliminate one (or both) by projecting.
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After projection
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• The combination of Cartesian product, selection and
projection allows you to bring together the related
information that was placed in different tables.
• This is the key operation in querying.
• It is called a join.
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RealPerson Table
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Credential Table
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Cartesian Product of Character and
Credential Tables (in Excel)
The credentials belong to Groening.
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Question
Do I throw out people even if I don’t have any
credentials for them?
There are different types of joins.
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References
 Database Systems, Rob and Coronel
 Database Systems, Connolly and Begg
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