VocaloidEX

Chap 2: Set Theory (Part 6)

FUNCTIONS:

A function f from a set X to a set Y is a relation from X to Y such that x Î X is related to one and only one y Î Y
X is called the domain & Y is called the range. We say x is mapped into y.
f = {(a, 3), (b, 3), (c, 5), (d, 1)}
Functions are described as:

Set of ordered pairs, example f as given before
Using a formula such as f(x) = expression, example : f (x) = 2x + 2
Illustration as in the example below

Chap 2: Set Theory (Part 5)

Relation between 2 sets:

™ A relation between two sets A and B is a subset of the Cartesian product AxB; A is called the source set and B is called the target set.

™ Often, we use notation aRb to denote that (a.b)єR and a~Rb to denote that (a,b)∉ ..

EXAMPLE:

Let U={0,1,2,3,4,5,6} represent set with bit strings

a) A= {2,4,5,6}

bit string is 0010111

b) B is the set of all odd integer, B Í U.

B= {1,3,5} the bit string is 0101010

c) C is a subset of U, containing all integers greater than 4.

C={5,6} the bit string is 0000011

COMPUTER REPRESENTATION OF SETS:
 There are many ways to represent sets using a computer.
 One of the method is to store elements using an arbitrary ordering of elements of the universal set.
◦ Ordered set (less time-consuming)
◦ Unordered set (time consuming)
 An arbitrary ordering of the elements of U (universal set), for instance, a1, a2, …, an, then representing a subset of A of U with bit string of length n, where the ith bit in this string is 1 if a1 belongs to A and is 0 if a does not belong to A.

EXAMPLE:
Let U={0,1,2,3,4,5,6} represent set with bit strings (binary form)
a) A= {2,4,5,6}
bit string is 0010111
b) B is the set of all odd integer, BÍ U.
B= {1,3,5} the bit string is 0101010
c) C is a subset of U, containing all integers greater than 4.
C={5,6} the bit string is 0000011