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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....}.





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