CS 3630 Database Design and Implementation
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Mathematical Relation
A mathematical relation is a subset of a Cartesian Product.
A1 A2 A3 … An = {(x1, x2, x3, …, xn): xi Ai}
R A1 A2 A3 … An
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Mapping and Relation
R1 = {(1, 2), (1, 5) , (2, 5)}
= {(x, y) | x is in A, y is in B, x < y and x <= 2}
A relationship between A and B
1
2
3
4
2
5
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How Many Possible Relations/Mappings between A and B
A = {1, 2, 3, 4}B = {2, 5}
R A B
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Functions
A function is a mapping
y = f(x)
x in a set Domain
y in a set Range
Mapping from Domain to Range
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Total or Partial
y = f(x) = x2
.1.22-24..
.1.44
4-416.
Total
Domain Real Numbers
Range Real Numbers
Partial
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Total or Partial
y = f(x) = x2
.1.22-24..
.1.44
4
16.
Total
Domain Real Numbers
RangeNon-negative Real Numbers
Total
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Functions
A function is a mapping
y = f(x)
x in a set Domain
y in a set Range
One-to-one?
One-to-Many?
Many-to-one?
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Functions
y = f(x)
For the same value of x, the value of y always the same.
y1 = f(3)
u = 2
y2 = f(u + 1)
Both y1 and y2 must have the same value!
It CANNOT be one-to-many!
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Functions
But for different values of x, the value of y could be the same.
y = f(x) = x2
f(2) = f(-2)
It’s not one-to-one!
It could be many-to-one.
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Relation Example
Student = {s: s is a student registered at UWP
this semester}
Course = {c: c is a course offered at UWP this
semester}
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Student Taking Course Student Course
S9
S2
S4
S5
C1
C2
C3
C5
Partial or Total?Partial or Total?
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ExampleS = {s | s is a student registered at UWP this semester}C = {c | c is a course offered at UWP this semester}
S C: All possible registrations
Registration = {(s, c): (s, c) S C and s is taking c}
Partial for both S and C
Many-to-many
It can be updated!
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TerminologyA = {1, 2, 3, 4}B = {2, 5}
R1 = {(1, 2), (1, 5), (2, 5)} R1 A B
Tuple: (1, 2), (1, 5) and (2, 5)Cardinality: 3Domain: A for the 1st component B for the 2nd component
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Using Tables for Relations
Att1 Att2 Att3
1 5 x
5 10 w
4 2 x
3 5 y
R = {(1, 5, x), (4, 2, x), (3, 5, y), (5, 10, w)}
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Database Relations (Tables)
Relation (Table)
Attribute (Field, column)
Tuple (Record, row)
Domain for each field
Degree (number of fields): 3
Cardinality (number of records): 4
Att1 Att2 Att3
1 5 x
5 10 w
4 2 x
3 5 y
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Table Instance
A subset of the Cartesian Product No duplicatesUpdate Tuples can be modified New tuples can be inserted Old tuples can be deletedDifferent instance after update
Student Course
S2 C2
S3 C2
S2 C5
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Relation Schema
A named relation defined by a set of attribute and domain pairs.
R (Att1:A1, Att2:A2, Att3:A3)
R (Att1, Att2, Att3)Domain: Att1: A1 Att2: A2 Att3: A3
The relation schema represents the Cartesian Product A1 A2 A3
Same table schema, different table instances.
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Relational Model
• What is a Relational Database?
A collection of relations (tables)
• Relational Database:
A collection of normalized relations (tables) with distinct relation/table names.
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Database Schema
Collection of Relation Schemas
Database definition
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Database Instance
Relation instances (Tables)
Actual records
Store Data
Update Data
Retrieve Data
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Properties of Relations
• Each relation has a distinct name (within the database)
• Each attribute has a distinct name (within the relation)
• Each tuple is distinct• Each cell contains exactly one atomic
(single) value• The order of attributes has no significance• The order of tuples has no significance
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Database UWP
Student(ID, Name, Phone, Address…)Course(ID, Title, Department, Credits…)Faculty(ID, Name, Department, Phone, Address…)Registration (Student, Course) Other possible fields?Schedule (Faculty, Course) Other possible fields?Other tables
Chapter 3
The Relational Model
3.2 Terminology
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