1 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

2 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

3 How Many Possible Relations/Mappings between A and B A = {1, 2, 3, 4} B = {2, 5} R  A  B

4 Functions Function is a mapping, but not one-to-one! 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 have the same value! It’s NOT one-to-many!

5 Functions (II) But for different values of x, the value of y could be the same. y = f(x) = x 2 f(2) = f(-2) It could be many-to-one.

6 Functions (III) Many-to-One Total / Partial (Total) X1 X2 X3 X4 A2 A4 A6 A8

7 Relation Example Student = {s: s is a student registered at UWP this semester} Course = {c: c is a course offered at UWP this semester}

8 Student Taking Course Student Course S9 S2 S4 S5 C1 C2 C3 C5 Partial or Total?

9 Student Taking Course Student Course S9 S2 S4 S5 C1 C2 C3 C5 Partial

10 Example S = {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!

11 Terminology R1 = {(1, 2), (1, 5), (2, 5)} R1  A  B Tuple: (1, 2), (1, 5) and (2, 5) Cardinality: 3 Domain: A for the 1 st component B for the 2 nd component

12 Using Tables for Relations Att1Att2 Att3 15x 510w 42x 35y R = {(1, 5, x), (4, 2, x), (3, 5, y), (5, 10, w)}

13 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 Att1Att2 Att3 15x 510w 42x 35y

14 Table Instance A subset of the Cartesian Product No duplicates Update Tuples can be modified New tuples can be inserted Old tuples can be deleted Different instance after update Student Course S2C2 S3C2 S2C5

15 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.

16 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.

17 Database Schema Collection of Relation Schemas Database definition

18 Database Instance Relation instances (Tables) Actual records Store Data Update Data Retrieve Data

19 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

20 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 21