Relational Math
CSC 240 (Blum)
1
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.
CSC 240 (Blum)
2
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
• Operations are called unary if they act on one
element and binary if they act on two elements.
– E.g square root is a unary algebraic operation.
– E.g. addition is a binary algebraic operation.
CSC 240 (Blum)
3
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.
CSC 240 (Blum)
4
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”
CSC 240 (Blum)
5
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 )
CSC 240 (Blum)
6
Selection Example: Actors Only (Design)
Customer.* refers to
all of the columns.
The condition (predicate)
selecting out particular rows.
CSC 240 (Blum)
7
Selection Example: PA Only (DataSheet)
CSC 240 (Blum)
8
Selection Example: PA Only (SQL)
Condition
CSC 240 (Blum)
9
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)
CSC 240 (Blum)
10
Added Some New Customers
CSC 240 (Blum)
11
Selection Example: Philadelphia AND
PA (Design)
CSC 240 (Blum)
ANDed conditions are
entered on the same line.
12
Selection Example: Philadelphia AND
PA (DataSheet)
CSC 240 (Blum)
13
Selection Example: Philadelphia AND
PA (SQL)
ANDed conditions
CSC 240 (Blum)
14
Selection Example: Customers from PA
or NJ (Design)
CSC 240 (Blum)
ORed conditions are
entered on separate
lines.
15
Selection Example: Customers from PA
or NJ (DataSheet)
CSC 240 (Blum)
16
Selection Example: Customers from PA
or NJ (SQL)
ORed Conditions
CSC 240 (Blum)
17
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 )
CSC 240 (Blum)
18
Projection Example: Customer’s first and
last names (Design)
Choose columns
and check to
show them.
CSC 240 (Blum)
19
Projection Example: Customer’s first and
last names (DataSheet)
CSC 240 (Blum)
20
Projection Example: Customer’s first and
last names (SQL)
Columns that will appear.
CSC 240 (Blum)
21
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 union-compatible.
• Then you can consider forming
– The union of the two sets
– The intersection of the two sets
– The set difference
CSC 240 (Blum)
22
A Simpsons Database
CSC 240 (Blum)
23
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)
CSC 240 (Blum)
24
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.
CSC 240 (Blum)
25
Union Example: Character and
RealPerson names (DataSheet)
CSC 240 (Blum)
26
Union Example: Character and
RealPerson names (DataSheet)
CSC 240 (Blum)
27
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!
CSC 240 (Blum)
28
Union Example: Character and
RealPerson names
CSC 240 (Blum)
29
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.
CSC 240 (Blum)
30
Intersection Example: People playing
themselves (DataSheet)
CSC 240 (Blum)
31
Intersection Example: People playing
themselves (Design, step 1)
CSC 240 (Blum)
32
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.
CSC 240 (Blum)
33
Intersection Example: People playing
themselves (SQL)
SQL has an INTERSECT operation like its UNION
operation, but it is not supported by Access.
CSC 240 (Blum)
34
Intersection Example: People playing
themselves (Design, version 2)
concatenation
subquery
Uses concatenation and a subquery.
CSC 240 (Blum)
35
Intersection Example: People playing
themselves (SQL, version 2)
Note: Access adds lots of parentheses.
CSC 240 (Blum)
36
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 unioncompatible tables.
CSC 240 (Blum)
37
Set Difference Example: Characters that
are not real people (DataSheet)
Ernest Borgnine and
James Brown removed.
CSC 240 (Blum)
38
Set Difference Example: Characters that
are not real people (Design)
Same as the second version of the
Intersection query except IN  NOT IN
CSC 240 (Blum)
39
Set Difference Example: Characters that
are not real people (SQL)
CSC 240 (Blum)
40
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.
CSC 240 (Blum)
41
How big is the Cartesian product?
Degree(B)
A
D
G
J
B
E
H
K
C
F
I
L
Cardinality(B)
Cardinality(A)
Degree(A)
A
G
D
D
J
N
O
U
E
V
Cardinality(AB) = Cardinality(A) * Cardinality(B)
Degree(AB) = Degree(A) + Degree(B)
CSC 240 (Blum)
42
CSC 240 (Blum)
A
B
C
A
N
A
B
C
G
O
A
B
C
D
U
A
B
C
D
E
A
B
C
J
V
D
E
F
A
N
D
E
F
G
O
D
E
F
D
U
D
E
F
D
E
D
E
F
J
V
G
H
I
A
N
G
H
I
G
O
G
H
I
D
U
G
H
I
D
E
G
H
I
J
V
J
K
L
A
N
J
K
L
G
O
J
K
L
D
U
J
K
L
D
E
J
K
L
J
V
43
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.
CSC 240 (Blum)
44
CSC 240 (Blum)
45
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.
CSC 240 (Blum)
46
A
B
C
A
N
A
B
C
G
O
A
B
C
D
U
A
B
C
D
E
A
B
C
J
V
D
E
F
A
N
D
E
F
G
O
D
E
F
D
U
D
E
F
D
E
D
E
F
J
V
G
H
I
A
N
G
H
I
G
O
G
H
I
D
U
G
H
I
D
E
G
H
I
J
V
J
K
L
A
N
J
K
L
G
O
J
K
L
D
U
J
K
L
D
E
J
K
L
J
V
CSC 240 (Blum)
47
match
After the selection
A
B
C
A
N
D
E
F
D
U
D
E
F
D
E
G
H
I
G
O
J
K
L
J
V
Now we have a table with two identical columns. We
can eliminate one (or both) by projecting.
CSC 240 (Blum)
48
After projection
A
B
C
N
D
E
F
U
D
E
F
E
G
H
I
O
J
K
L
V
• 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.
CSC 240 (Blum)
49
RealPerson Table
CSC 240 (Blum)
50
Credential Table
CSC 240 (Blum)
51
Cartesian Product of Character and
Credential Tables (in Excel)
The credentials belong to Groening.
CSC 240 (Blum)
52
Question
Do I throw out people even if I don’t have any
credentials for them?
There are different types of joins.
CSC 240 (Blum)
53
References
• Database Systems, Rob and Coronel
• Database Systems, Connolly and Begg
CSC 240 (Blum)
54