Page 8 - DMTH401_REAL ANALYSIS

Real Analysis

Notes ‘Real Analysis’ is an important branch of Mathematics which mainly deals with the study of real
numbers. What is, then, the system of the real numbers? We shall try to find an answer to this
question as well as some other related questions in this unit. Also, we shall give the geometrical
representation of the real numbers.

1.1 Sets and Functions

As you all know modern Mathematics is based on the ideas that are expressed in the language of
sets and functions. Here you set knowledge of certain basic concepts of Set Theory which are
quite familiar to you. These concepts will also serve an important purpose of recalling certain
notations and terms that will be used throughout our discussion with you.

1.1.1 Sets

As you are used to the phrases like the ‘team’ of cricket players, the ‘army’ of a country, the
‘committee’ on the education policy, the ‘panchayat’ of a village, etc. The terms ‘team’, ‘army’,
‘committee’, panchayat’, etc., indicate the notion of a ‘collection’ or ‘totality’ or ‘aggregate’ of
objects. These are well-known examples of a set.
Therefore, our starting point is an informal description of the term ‘set’. A set is treated as an
undefined term just as a point in Geometry is undefined. However, for our purpose we say that
a set is a well-defined collection of objects. A collection os well-defined of it is possible to say
whether a given object belongs to the collection or not.

The following are some examples of sets:
1. The collection of the students registered in Excel Books.
2. The collection of the planets namely Jupiter, Saturn, Earth, Pluto, Venus, Mercury, Mars,
Uranus and Neptune.
3. The collection of all the countries in the world. (Do you know how many countries are
there in the world?)

4. The collection of numbers, 1, 2, 3, 4, ……
If we consider the collection of tall persons or beautiful ladies or popular leaders, then these
collections are not well-defined and hence none of them forms a set. The reason is that the words
‘tal’ ‘beautiful’ or ‘popular’ are not well-defined. The objects constituting a set are called its
elements or members or points of the set. Generally, sets are denoted by the capital letters A, B,
C etc. and the elements are denoted by the small letters a, b, c etc. If S is any set and x is an
element of S, we express it by writing that x  S, where the symbol means ‘belongs to’ or ‘is a
member of’. If x is not an element an element of a set S, we write xS. For example , if S is the set
containing 1, 2, 3, 4 only, then 2 S and 5  S.

You know that there are two method of describing a set. One is known as the Tabular method
and the other is the Set-Builder method. In the tabular method we describe a set by actually
listing all the elements belonging to it.

Example: If S is the set consisting of all small letters of English alphabet, then we write
S = {a, b, c,...,x, y, z}.
If N is the set of all natural numbers, then we write
N = {1, 2, 3....}.

2 LOVELY PROFESSIONAL UNIVERSITY