When nothing is everything in Set theory

In an earlier post, I discussed the basic and most important aspects of Set theory, Functions and Real Number System. In the same, there was a significant discussion about the union and intersection of sets. Restating the facts again, given a collection $ \mathcal{A}$ of sets, the union of the elements of $ \mathcal{A}$ is defined by $ \displaystyle{\bigcup_{A \in \mathcal{A}}} A := {x : x \in A \textrm{ for at least one } A \in \mathcal{A} }$ . The…

Real Sequences

Sequence of real numbers A sequence of real numbers (or a real sequence) is defined as a function $ f: \mathbb{N} \to \mathbb{R}$ , where $ \mathbb{N}$ is the set of natural numbers and $ \mathbb{R}$ is the set of real numbers. Thus, $ f(n)=r_n, \ n \in \mathbb{N}, \ r_n \in \mathbb{R}$ is a function which produces a sequence of real numbers $ r_n$ . It's customary to write a sequence as form of functions in brackets, e.g.; $…

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:

Interesting Egyptian Fraction Problem

Here is an interesting mathematical puzzle alike problem involving the use of Egyptian fractions, whose solution sufficiently uses the basic algebra. Problem Let a, b, c, d and e be five non-zero complex numbers, and; $ a + b + c + d + e = -1$ ... (i) $ a^2+b^2+c^2+d^2+e^2=15$ ...(ii) $ \dfrac{1}{a} + \dfrac{1}{b} +\dfrac{1}{c} +\dfrac{1}{d} +\dfrac{1}{e}= -1$ ...(iii) $ \dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}+\dfrac{1}{d^2}+\dfrac{1}{e^2}=15$ ...(iv) $ abcde = -1 $ ...(v). Solve and find the values of a, b, c, d…

Euler’s (Prime to) Prime Generating Equation

The greatest number theorist in mathematical universe, Leonhard Euler had discovered some formulas and relations in number theory, which were based on practices and were correct to limited extent but still stun the mathematicians. The prime generating equation by Euler is a very specific binomial equation on prime numbers and yields more primes than any other relations out there in number theory. Euler told that the equation $ f(x)=x^2+x+k$ yields many prime numbers with the values of x being input…

Irrational Numbers Featured Image

Irrational Numbers and The Proofs of their Irrationality

"Irrational numbers are those real numbers which are not rational numbers!" Def.1: Rational Number A rational number is a real number which can be expressed in the form of where $ a$ and $ b$ are both integers relatively prime to each other and $ b$ being non-zero. Following two statements are equivalent to the definition 1. 1. $ x=\frac{a}{b}$ is rational if and only if $ a$ and $ b$ are integers relatively prime to each other and $…

On Ramanujan’s Nested Radicals

Ramanujan (1887-1920) discovered some formulas on algebraic nested radicals. This article is based on one of those formulas. The main aim of this article is to discuss and derive them intuitively. Nested radicals have many applications in Number Theory as well as in Numerical Methods . The simple binomial theorem of degree 2 can be written as: $ {(x+a)}^2=x^2+2xa+a^2 \ \ldots (1)$ Replacing $ a$ by $ (n+a)$ where $ x, n, a \in \mathbb{R}$ , we can have $…

Free Online Algebra and Topology Books

This is a brief list of free e-books on Algebra, Topology and Related Mathematics. I hope it will be very helpful to all students and teachers searching for high quality content. If any link is broken, please email me at gaurav(at)gauravtiwari.org. Abstract Algebra OnLine by Prof. Beachy This site contains many of the definitions and theorems from the area of mathematics generally called abstract algebra. It is intended for undergraduate students taking an abstract algebra class at the junior/senior level,…