Page 9 - DMTH401_REAL ANALYSIS

Unit 1: Sets and Numbers

This is also called an explicit representation of a set. Notes
In the set-builder method, a set is described by stating the property which determines the set as
a well-defined collection. Suppose p denotes this property and x is an element of a set S. Then

S = {x: x satisfies p).

Example: The two sets S and N can be written as
S = {x: x is a small letter of English alphabet}
N = {n: n is a natural number).
This is also called an implicit representation of a set.
Note that in the representation of sets, the elements of a set are not repeated. Also, the elements
may be listed in any manner.

Example: Write the set S whose elements are all natural numbers between 7 and 12
including both 7 and 12 in the tabular as well as in the set-builder forms.
Solution: Tabular form is S = {7, 8, 9, 10, 11, 12, }.

Set-builder form is S = { n  N: 7  n  12, }.
The following standard notations are used for the sets of numbers:
N = Set of all natural numbers
= {1, 2, 3....}
= {n:n is a natural number)

= Set of all positive integers.
Z = Set of all integers
= { ....–3, –2, –1, 0, 1, 2, 3, ....}
= {p:p is an integer).

Q = Set of all rational numbers
P
= {x : x = , p Z, q  Z, q  0).
q
R = Set of real numbers
= {x : x is a real number).
We shall, however, discuss the development of the system of real numbers.

A set is said to be finite if it has a finite number of elements. A set is said to be infinite if it is not
finite. We shall, however, give a mathematical definition of finite and infinite sets in Unit 2.
Note that an element of a set must be carefully distinguished from the set consisting of this
element. Thus, for instance, you must distinguish
x, {x}, {{x}}
from each other

We talk of equality of numbers, equality of objects, etc.
The question, therefore, arises: What is the notion of the equality of sets?

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