Set Theory, Functions and Real Number System

SETS

In mathematics, Set is a well defined collection of distinct objects. The theory of Set as a mathematical discipline rose up with George Cantor, German mathematician, when he was working on some problems in Trigonometric series and series of real numbers, after he recognized the importance of some distinct collections and intervals.

Cantor defined the set as a ‘plurality conceived as a unity’ (many in one; in other words, mentally putting together a number of things and assigning them into one box).

Mathematically, a Set $ S$ is ‘any collection’ of definite, distinguishable objects of our universe, conceived as a whole. The objects (or things) are called the elements or members of the set $ S$ . Some sets which are often pronounced in real life are, words like ”bunch”, ”herd”, ”flock” etc. The set is a different entity from any of its members.

For example, a flock of birds (set) is not just only a single bird (member of the set). ‘Flock’ is just a mathematical concept with no material existence but ‘Bird’ or ‘birds’ are real.

Representing sets

Sets are represented in two main ways:

Equations- A Basic Introduction

Applied mathematics is one which is used in day-to-day life, in solving tensions (problems) or in business purposes. Let me write an example: George had some money. He gave 14 Dollars to Matthew. Now he has 27 dollars. How much money had he? If you are familiar with day-to-day calculations --you must say that George had 41 dollars, and since he had 41, gave 14 to Matthew saving 27 dollars. That's right? Off course! This is a general(layman) approach. 'How…

Numbers – The Basic Introduction

If mathematics was a language, logic was the grammar, numbers should have been the alphabet. There are many types of numbers we use in mathematics, but at a broader aspect we may categorize them in two categories: 1. Countable Numbers 2. Uncountable Numbers The numbers which can be counted in nature are called Countable Numbers and the numbers which can not be counted are called Uncountable Numbers. Well, this is not the correct way to classify the bunch of types…

Mathematical Logic – The basic introduction

What is Logic? If mathematics is regarded as a language, then logic is its grammar. In other words, logical precision has the same importance in mathematics as grammatical accuracy in a language. As linguistic grammar has sentences, statements--- logic has them too. After we discuss about Sentence & Statements, we will proceed to further logical theories . Sentences & Statements A sentence is a collection of some words, those together having some sense. For example: Math is a tough subject.…