devotion & mathematics

                        In set theory,  there are two methods of representing a set :  1. Roaster or Tabular form  2. Set- Builder form Roaster Form :           In roaster form , all the elements of a set are listed, the elements are being                         separated by commas  and are enclosed within braces { } .  For example  1. The set of all even positive integers lessthan 7 is described in roaster form as{ 2,4,6 }.  2. The set of all natural numbers which divide 42 is { 1,2,3,6,7,14,21,42 }  3. The set of all vowels in the English alphabet is { a,e,i,o,u } etc. Note :   In roaster form , the order in which the elements are listed is immaterial. i.e. in the        example 2, the  set in the example 2 can also be written as { 1,7,2,3,14,21,42,6 }  S...

                                    A well- defined collection of objects in the sense that we can definitely  decide   whether a given  particular object belongs to a given collection  or not.  For example :   * The collection of numbers 2,4,6,8,... is  well-defined because these numbers are formed      by     multiply natural numbers with 2.   * The collection of five most renowed mathematicians of the world is not well-defined ,       because   the criterion for determining a mathematician as most renowed may vary from      person to person.   If you have a question in your mind is that  " Is  " Collection "  a synonym to " Set " " then    answer is " not " .  Now we see the difference between a collection and a set.   " Collec...

  OPEN & CLOSED SETS In this session we see open & closed sets with their properties. OPEN SET : -      A non empty set A in the real line R is called an open set , every point in A is an interior point of A.                                                                     Examples:- * (a, b)  is an open set.  * [a, b] is not open since the end points a, b are not interior points. * Every finite set is not open. * The sets N, Z and Q are not open sets. * The set R of all real numbers is open. * The set of complex numbers |z|<1 is open. * The set of complex numbers |z|  ≤1 is not open. * Every neighbourhood is an open set. Properties of open sets:- 1. Every open set is a union of open intervals. 2. Union of two open sets is open. 3. Intersection of two open sets is ...

In mathetics, the terms the neighborhood of a point, interior and limit points have basic role to define open and closed sets. Before going to open and closed sets , here we see the concepts of neighborhood , interior and limit point.   ** N for set of natural numbers   ** Z is for set of integers   ** Q is for set of rational numbers   ** R is for set of real numbers NEIGHBOURHOOD OF A POINT Let  x be a point of a nonempty subset A of the real line R.  The neighborhood of x with radius ϵ is denoted by Nϵ(x) and is given by Nϵ(x) = {y Ꜫ A/|x-y|<r}  The neighborhood of x is also simply written as nbd of x or nbd(x). Also the neighborhood of x, Nϵ(x) is also called as ϵ-nbd of x. Note :      1. An open interval (a,b) is a nbd of its points.      2. A closed interval [a,b] is a nbd of its  points except for end points a and b.      3. The sets N,Z,Q are not nbd's of its points.      4...

  BASIC DEFINITIONS :                                      In this session , I am going to give definitions and few examples of   * set  * upper bound of a set  * lower bound of a set  * greatest lower bound or infimum of a set  * least upper bound or supremum of a set  * bounded above, bounded below and  bounded sets.  * Diameter of a set.  * Distance between a point and a set   SET :         A set is a collection of numbers or elements or objects etc.       For example :               * the collection of numbers which contain 1,2,3,... is called the set of natural numbers.              * The  collection of all people aged between 25-30 is called a set.              * The colle...