Home » 2011 » November

# If you are doing good, go for great! And if great    -  chase the best.

This is a statement which I follow the most. Time is changing rapidly for me and I am more rapidly. At MY DIGITAL NOTEBOOK —I devoted my heart and brain to mathematical and recreational writing. But, wpgaurav.wordpress.com looked ugly in the sense of cuteness of my blog.
A custom-domain is not very important in the process of getting readers —your blogging does so, but it is very important if you are serious to your name. I love my name and that’s why I opted for my own domain than traditional *.wordpress.com URL. I was looking for an excellent domain name for my blog and what on the earth could be better than my full name? I didn’t look for gauravtiwari.com, whether that was available or not and directly went to register gauravtiwari.org. This made a sense because I treat myself as an organisation and I successfully got it registered. The process was extremely easy and effective, however I failed purchasing the domain twice. Reason —both of my cards weren’t international. I used my friend’s VISA card and finally in third attempt —I had it (well, I paid him back in cash).
Some people might think that I was inspired by declaration of WordAds on WordPress.com but that is not true. When I reached at 40000 hits, I dreamt for my domain –which will have no range–Full of recreations—full of knowledge. I don’t have more to write on this topic since I am completing my book ‘A Trip To Mathematics‘. If you are a regular reader of this blog please update you bookmarks. wpgaurav.wordpress.com is now gauravtiwari.org Thanks for your support my friends and WordPress.com guys! I love math. I love WordPress. I love you.

## WordPress URL Shortening: An Interesting Consequence

Long post and page URLs can be annoying. There are times when you may want to edit WordPress slugs for long titled posts or pages, so you can share a shorter link with friends. Here’s an example: http://wpgaurav.wordpress.com/2011/02/06/derivative-of-x-squared-is-2x-or-x-where-is-the-fallacy/ (one of the most popular posts from my blog). Would you like to write these (about) hundred characters? Or would you like to use WP.ME link shortener, which for this post was http://wp.me/p14tlY-hv ? Well, second idea sounds better than first one. But you’ll never get the same satisfaction from it because blog URL http://wpgaurav.wordpress.com is not visible, and also looks odd when typed in address bar. What then? This post is on some choices you might want to try. But before we go, let me explain what slug is?
A slug is a few words that describe a post or a page. Slugs are usually an URL friendly version of the post title (which has been automatically generated by WordPress), but a slug can be anything you like. Slugs are meant to be used with permalinks as they help describe what the content at the URL is. The post slug is the part of the URL after the date in a post’s URL. When the default post slug is created, all letters will be converted to lowercase, spaces will be exchanged with dashes, and any special characters will be removed. You can modify the post slug by clicking the Edit button next to it in visual editor. When you’re finished editing, click Save and then Update Post/Page. »via WordPress Codex

# How to shorten a WordPress post or page slug with a more efficient way?

Here are some interesting ways to define a personalised, very short and share-friendly version of URL for your post.

1. Eliminate the ‘time-stamp’ from the slug and
http://wpgaurav.wordpress.com/derivative-of-x-squared-is-2x-or-x-where-is-the-fallacy/
redirects to the same post at
http://wpgaurav.wordpress.com/2011/02/06/derivative-of-x-squared-is-2x-or-x-where-is-the-fallacy/
So it got a little shorter.
2. Want it to be even shorter? Then cut one or more characters from the URL: For example:
http://wpgaurav.wordpress.com/2011/02/06/derivative-of-x-squared-is-2x-or-x-where-is-the-fallacy/
http://wpgaurav.wordpress.com/2011/02/06/derivative-of-x-squared-is-2x-or-x-where-is/
http://wpgaurav.wordpress.com/derivative-of-x-squared-is-2x-or-x/
etc.  are equivalent and redirect to the same post.
3. Want it be shorter still? Then eliminate everything but single word. e.g., http://wpgaurav.wordpress.com/derivative/
Looks cool, no? Yes, Just like a Page URL on WordPress.com! And it redirects to the same post as all other links above.
4. Is the word too long for you? Then abbreviate it to
http://wpgaurav.wordpress.com/der/
5. Want it be still shorter? Try the shortest, a single letter: http://wpgaurav.wordpress.com/d/ ! Well, it does not redirect to http://wpgaurav.wordpress.com/2011/02/06/derivative-of-x-squared-is-2x-or-x-where-is-the-fallacy/
Where does it go? It goes to d’ Alembert’s Test of Convergence . Reason is specified below at point 2.

Here are some points, which follow and guide you on shortening URLs:

1. Hyphens are not important when considering shortened-urls.
2. Any URL of the type http://wpgaurav.wordpress.com/d redirects to a post/page which slug starts with ‘d’. If there are two posts having slugs starting with the same letter of the alphabet (here d),then it will, in alpahabetical order, redirect to that post which first word (if same then second word) comes first in English Dictionary. For example http://wpgaurav.wordpress.com/d will redirect to http://wpgaurav.wordpress.com/d-alembert/ rather than to http://wpgaurav.wordpress.com/derivative/
3. If two/more posts have the same slugs and you are using an identical shortener that it might go to any of them, then that will (should) redirect to the post which was published earlier.
4. Some slugs are not allowedin posts or pages on WordPress.com and they are:
• Periods(.) are not allowed in slugs.
• /activate/
• /category/
• /feed/
• /i/
• /next/
• /signup/
• /tag/
• /wp-content/
5. You can not shorten your post URLs /page URLs (this way) until published.
6. This trick also works for self hosted WordPress.org blogs.

So, are you going to experiment with your WordPress.com Post URLs?

Disclaimer: There is no technical basis for these shortening tricks. They are soley based on experiments with WordPress.com slugs. This post was made under supervision of timethief, I am grateful to her. I would also like to say thanks to Ganesh Dhamodkar, who was the guy I tested all these with.
Feel free to comment if you are getting any problem in shortening URLs or if just want to say ‘Hi’.

Long URL for this Post: http://wpgaurav.wordpress.com/2011/11/24/wordpress-url-shortening/

Short URL for this post: http://wpgaurav.wordpress.com/wor/

## A Trip to Mathematics: Part IV Numbers

If logic is the language of mathematics, Numbers are the alphabet. There are many kinds of number we use in mathematics, but at a broader aspect we may categorize them in two categories:
1. Countable Numbers
2. Uncountable Numbers
The names are enough to explain the properties of above 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 of numbers. We have some formal names for special types of numbers, like Real numbers, Complex Numbers, Rational Numbers, Irrational Numbers etc.. We shall discuss these non-interesting numbers (let me say them non-interesting) at first and then some interesting numbers(those numbers are really interesting to learn). Although in this post I have concisely described the classification, I will rigorously discuss them later.
Let me start this discussion with the memorable quote by Leopold Kronecker:

“God created the natural numbers, and all the rest is the work of man.”

Actually, he meant to say that all numbers, like Real Numbers, Complex Numbers, Fractions, Integers, Non-integers etc. are made up of the numbers given by God to the human. These God Gifted numbers are actually called Natural Numbers. Natural Numbers are the numbers which are used to count things in nature.

Eight pens, Eighteen trees, Three Thousands people etc. are measure of natural things and thus ‘Eight’, ‘Eighteen’, ‘Three Thousands’ etc. are called natural numbers and we represent them numerically as ’8′, ’18′, ’3000′ respectively. So, if 8, 18, 3000 are used in counting natural things, are natural numbers. Similarly, 1, 2, 3, 4, and other numbers are also used in counting things —thus these are also Natural Numbers.

Let we try to form a set of Natural Numbers. What will we include in this set?

1?                    (yes!).
2?                    (yes).
3?                     (yes).
….
1785?                (yes)
…and          so on.

This way, after including all elements we get a set of natural numbers {1, 2, 3, 4, 5, …1785, …, 2011,….}. This set includes infinite number of elements. We represent this set by Borbouki’s capital letter N, which looks like $\mathbb{N}$ or bold capital letter N ($\mathbf{N}$ where N stands for NATURAL. We will define the set of all natural numbers as:

$\mathbb{N} := \{ 1, 2, 3, 4, \ldots, n \ldots \}$.

It is clear from above set-theoretic notation that $n$-th element of the set of natural numbers is $n$.
In general, if a number $n$ is a natural number, we right that $n \in \mathbb{N}$.
Please note that some mathematicians (and Wolfram Research) treat ’0′ as a natural number and state the set as $\mathbb{N} :=\{0, 1, 2, \ldots, n-1, \ldots \}$, where $n-1$ is the nth element of the set of natural numbers; but we will use first notion since it is broadly accepted.

Now we shall try to define Integers in form of natural numbers, as Kronecker’s quote demands. Integers (or Whole numbers) are the numbers which may be either positives or negatives of natural numbers including 0.
Few examples are 1, -1, 8, 0, -37, 5943 etc.
The set of integers is denoted by $\mathbb{Z}$ or $\mathbf{Z}$ (here Z stands for ‘Zahlen‘, the German alternative of integers). It is defined by
$\mathbb{Z} := \{ \pm n: n \in \mathbb{N} \} \cup \{0\}$
i.e., $\mathbb{Z} := \{\ldots -3, -2, -1, 0, 1, 2, 3 \ldots \}$.

Now, if we again consider the statement of Kronecker, we might ask that how could we prepare the integer set $\mathbb{Z}$ by the set $\mathbb{N}$ of natural numbers? The construction of $\mathbb{Z}$ from $\mathbb{N}$ is motivated from the requirement that every integer can be expressed as difference of two positive integers (i.e., Natural Numbers). Let $a,b,c,d \in \mathbb{N}$ and a relation ρ is defined on $\mathbb{N} \times \mathbb{N}$ by $(a,b) \rho (c,d)$ if and only if $a+d = b+c$. The relation ρ is an equivalence relation and the equivalence classes under ρ are called integers and defined as $\mathbb{Z} := \mathbb{N} \times \mathbb{N} /\rho$. Now we can define set of integers by an easier way, as $\mathbb{Z}:= \{a-b; \ a,b \in \mathbb{N}\}$. Thus an integer is a number which can be produced by difference of two or more natural numbers. And similarly as converse defintion, positive integers are called Natural Numbers.
After Integers, we head to rational numbers. Say it again– ‘ratio-nal numbers‘ –numbers of ratio.

Image via Wikipedia

A rational number $\frac{p}{q}$ is defined as a ratio of an integer p and a non-zero integer q. (Well that is not a perfect definition, but as an introduction it is great for understanding.) The set of rational numbers is defined by $\mathbb{Q}$.
Once integers are formed, we can form Rational (and Irrational numbers: numbers which are not rational ) using integers.
We consider an ordered pair $(p,q):=\mathbb{Z} \times (\mathbb{Z} \setminus \{0 \})$ and another ordered pair $(r,s):=\mathbb{Z} \times (\mathbb{Z} \setminus \{0\})$ and define a relation ρ by $(p,q) \rho (r,s) \iff ps=qr$ for $p,q,r,s \in \mathbb{Z}, \ q, r \ne 0$. Then ρ is an equivalence relation of rationality, class (p,q). The set $\mathbb{Z} \times (\mathbb{Z} \setminus \{0\})/\rho$ is denoted by $\mathbb{Q}$ (and the elements of this set are called rational numbers).
In practical understandings, the ratio of integers is a phrase which will always help you to define the rational numbers. Examples are $\frac{6}{19}, \ \frac{-1}{2}=\frac{-7}{14}, \ 3\frac{2}{3}, \ 5=\frac{5}{1} \ldots$. Set of rational numbers includes Natural Numbers and Integers as subsets.
Consequently, irrational numbers are those numbers which can not be represented as the ratio of two integers. For example $\pi, \sqrt{3}, e, \sqrt{11}$ are irrationals.
The set of Real Numbers is a relatively larger set, including the sets of Rational and Irrational Numbers as subsets. Numbers which exist in real and thus can be represented on a number line are called real numbers. As we formed Integers from Natural Numbers; Rational Numbers from Integers, we’ll form the Real numbers by Rational numbers.
The construction of set $\mathbb{R}$ of real numbers from $\mathbb{Q}$ is motivated by the requirement that every real number is uniquely determined by the set of rational numbers less than it. A subset $L$ of $\mathbb{Q}$ is a real number if L is non-empty, bounded above, has no maximum element and has the property that for all $x, y \in \mathbb{Q}, x < y$ and $y \in L$ implies that $x \in L$. Real numbers are the base of Real Analysis and detail study about them is case of study of Real Anlaysis.
Examples of real numbers include both Rational (which also contains integers) and Irrational Numbers.

The square root of a negative number is undefined in one dimensional number line (which includes real numbers only) and is treated to be imaginary. The numbers containing or not containing an imaginary number are called complex numbers.
Some very familiar examples are $3+\sqrt{-1}, \sqrt{-1} =i, \ i^i$ etc. We should assume that every number (in lay approach) is an element of a complex number. The set of complex numbers is denoted by $\mathbb{C}$. In constructive approach, a complex number is defined as an ordered pair of real numbers, i.e., an element of $\mathbb{R} \times \mathbb{R}$ [i.e., $\mathbb{R}^2$] and the set as $\mathbb{C} :=\{a+ib; \ a,b \in \mathbb{R}$. Complex numbers will be discussed in Complex Analysis more debately.
We verified Kronecker’s quote and shew that every number is sub-product of postive integers (natural numbers) as we formed Complex Numbers from Real Numbers; Real Numbers from Rational Numbers; Rational Numbers from Integers and Integers from Natural Numbers. //
Now we reach to explore some interesting kind of numbers. There are millions in name but few are the follow:
Even Numbers: Even numbers are those integers which are integral multiple of 2. $0, \pm 2, \pm 4, \pm 6 \ldots \pm 2n \ldots$ are even numbers.

Odd Numbers: Odd numbers are those integers which are not integrally divisible by 2. $\pm 1, \pm 3, \pm 5 \ldots \pm (2n+1) \ldots$ are all odd numbers.

Prime Numbers: Any number $p$ greater than 1 is called a prime number if and only if its positive factors are 1 and the number $p$ itself.
In other words, numbers which are completely divisible by either 1 or themselves only are called prime numbers. $2, 3, 5, 7, 11, 13, 17, 19, 23, 29 \ldots$ etc. are prime numbers or Primes. The numbers greater than 1, which are not prime are called Composite numbers.
Twin Primes: Consecutive prime numbers differing by 2 are called twin primes. For example 5,7; 11,13; 17,19; 29,31; … are twin primes.

Pseudoprimes: Chinese mathematicians claimed thousands years ago that a number $n$ is prime if and only if it divides $2^n -2$. In fact this conjecture is true for $n \le 340$ and false for upper numbers because first successor to 340, 341 is not a prime ($31 \times 11$) but it divides $2^{341}-2$. This kind of numbers are now called Pseudoprimes. Thus, if n is not a prime (composite) then it is pseudoprime $\iff n | 2^n-2$ (read as ‘n divides 2 powered n minus 2‘). There are infinitely many pseudoprimes including 341, 561, 645, 1105.

Carmichael Numbers or Absolute Pseudoprimes: There exists some pseudoprimes that are pseudoprime to every base $a$, i.e., $n | a^n -a$ for all integers $a$. The first Carmichael number is 561. Others are 1105, 2821, 15841, 16046641 etc.

e-Primes: An even positive integer is called an e-prime if it is not the product of two other even integers. Thus 2, 6, 10, 14 …etc. are e-primes.

Germain Primes: An odd prime p such that 2p+1 is also a prime is called a Germain Prime. For example, 3 is a Germain Prime since $2\times 3 +1=7$ is also a prime.
Relatively Prime: Two numbers are called relatively prime if and only their greatest common divisor is 1. In other words, if two numbers are such that no integer, except 1, is common between them when factorizing. For example: 7 and 9 are relatively primes and same are 15, 49.

Perfect Numbers: A positive integer n is said to be perfect if n equals to the sum of all its positive divisors, excluding n itself. For example 6 is a perfect number because its divisors are 1, 2, 3 and 6 and it is obvious that 1+2+3=6. Similarly 28 is a perfect number having 1, 2, 4, 7, 14 (and 28) as its divisors such that 1+2+4+7+14=28. Consecutive perfect numbers are 6, 28, 496, 8128, 33550336, 8589869056 etc.

Mersenne Numbers and Mersenne Primes: Numbers of type $M_n=2^n-1; \ n \ge 1$ are called Mersenne Numbers and those Mersenne Numbers which happen to be Prime are called Mersenne Primes. Consecutive Mersenne numbers are 1, 3 (prime), 7(prime), 15, 31(prime), 63, 127.. etc.

Catalan Numbers: The Catalan mumbers, defined by $C_n = \dfrac{1}{n+1} \binom{2n}{n} = \dfrac{(2n)!}{n! (n+1)!} \ n =0, 1, 2, 3 \ldots$ form the sequence of numbers 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, …

Triangular Number: A number of form $\dfrac{n(n+1)}{2} \ n \in \mathbb{N}$ represents a number which is the sum of n consecutive integers, beginning with 1. This kind of number is called a Triangular number. Examples of triangular numbers are 1 (1), 3 (1+2), 6 (1+2+3), 10(1+2+3+4), 15(1+2+3+4+5) …etc.

Square Number: A number of form $n^2 \ n \in \mathbb{N}$ is called a sqaure number.
For example 1 ($1^2$), 4 ($2^2$), 9($3^2$), 16 ($4^2$)..etc are Square Numbers.

Palindrome: A palindrome or palindromic number is a number that reads the same backwards as forwards. For example, 121 is read same when read from left to right or right to left. Thus 121 is a palindrome. Other examples of palindromes are 343, 521125, 999999 etc.

//

## How Many Fishes in One Year? [A Puzzle in Making]

This is a puzzle which I told to my classmates during a talk, a few days before. I did not represent it as a puzzle, instead of a talk suggesting the importance of Math in general life. This is partially solved for me and I hope you will run your brain-horse to help me solve it completely. If you didn’t notice, this puzzle is not a part of A Trip To Mathematics series. Puzzle which I discussed in the talk was something like this:

“Let I have seven fishes in a huge tank of water —four male and three females. Those were allowed to sex independently but under some conditions. One male is allowed to have intercourse with female, unless other has done so. A male can have intercourse with any number of female fishes possible. If we assume that first female fish could give 100 eggs, second female fish could give 110, third 90. We are also known that a female fish might lay eggs in 21 days since the date of sex with male. The children fish are reproductive only after those are 30 days old. In each bunch of children fish of individual female fish, 60% die. In remaining 40% child fishes, the ratio of male and female is 3:2. If two fishes (male and female) can not do intercourse with each other if those are born to same mother fish, one male is allowed to have intercourse with female, unless other has done so, a male can have intercourse with any number of fishes possible, every three female fishes lay eggs in order of 100,110 and 90 eggs out of which only 40% remain alive having a ratio of male and females of 1:1 and the same rule applies to third, fourth and consecutive generations of fishes; then find the number of total fishes in my tank after one year (365 days).”

I have done many proofreads of this puzzle and found it valid. Your comments, your ideas and suggestions might help me working more rigorously on this puzzle. This puzzle is neither too hard nor too easy. I will be updating this post frequently as my work on this puzzle is directed towards a correct way.

## A Trip to Mathematics: Part III Relations and Functions

‘Michelle is the wife of Barak Obama.’
‘John is the brother of Nick.’
‘Robert is the father of Marry.’
‘Ram is older than Laxman.’
‘Mac is the product of Apple Inc.’
After reading these statements, you will realize that first ‘Noun’ of each sentence is some how related to other. We say that each one noun is in a RELATIONSHIP to other. Mischell is related to Barak Obama, as wife. John is related to Nick, as brother. Robert is related to Marry, as father. Ram is related to Laxman in terms of age(seniority). Mac is related to Apple Inc. as a product.These relations are also used in Mathematics, but a little variations; like ‘alphabets’ or ‘numbers’ are used at place of some noun and mathematical relations are used between them. Some good examples of relations are:

is less than
is greater than
is equal to
is an element of
belongs to
divides
etc. etc.

Some examples of regular mathematical statements which we encounter daily are:

4<6 : 4 is less than 6.
5=5 : 5 is equal to 5.
6/3 : 3 divides 6.

For a general use, we can represent a statement as:
”some x is related to y”
Here ‘is related to’ phrase is nothing but a particular mathematical relation. For mathematical convenience, we write ”x is related to y” as $x \rho y$. x and y are two objects in a certain order and they can also be used as ordered pairs (x,y).
$(x,y) \in \rho$ and $x \rho y$ are the same and will be treated as the same term in further readings. If $\rho$ represents the relation motherhood, then $\mathrm {(Jane, \ John)} \in \rho$ means that Jane is mother of John.
All the relations we discussed above, were in between two objects (x,y), thus they are called Binary Relations. $(x,y) \in \rho \Rightarrow \rho$ is a binary relation between a and b. Similarly, $(x,y,z) \in \rho \Rightarrow \rho$ is a ternary (3-nary) relation on ordered pair (x,y,z). In general a relation working on an n-tuple $(x_1, x_2, \ldots x_n) \in \rho \Rightarrow \rho$ is an n-ary relation working on n-tuple.
We shall now discuss Binary Relations more rigorously, since they have solid importance in process of defining functions and also in higher studies. In a binary relation, $(x,y) \in \rho$, the first object of the ordered pair is called the the domain of relation ρ and is defined by
$D_{\rho} := \{x| \mathrm{for \ some \ y, \ (x,y) \in \rho} \}$ and also the second object is called the range of the relation ρ and is defined by $R_{\rho} := \{y| \mathrm{for \ some \ y, \ (x,y) \in \rho} \}$.
There is one more thing to discuss about relations and that is about equivalence relation.
A relation is equivalence if it satisfies three properties, Symmetric, Reflexive and Transitive.
I mean to say that if a relation is symmetric, reflexive and transitive then the relation is equivalence. You might be thinking that what these terms (symmetric, reflexive and transitive) mean here. Let me explain them separately:
A relation is symmetric: Consider three sentences “Jen is the mother of John.”; “John is brother of Nick.” and “Jen, John and Nick live in a room altogether.”
In first sentence Jen has a relationship of motherhood to John. But can John have the same relation to Jen? Can John be mother of Jen? The answer is obviously NO! This type of relations are not symmetric. Now consider second statement. John has a brotherhood relationship with Nick. But can Nick have the same relation to John? Can Nick be brother of John? The answer is simply YES! Thus, both the sentences “John is the brother of Nick.” and “Nick is the brother of John.” are the same. We may say that both are symmetric sentences. And here the relation of ‘brotherhood’ is symmetric in nature. Again LIVING WITH is also symmetric (it’s your take to understand how?).
Now let we try to write above short discussion in general and mathematical forms. Let X and Y be two objects (numbers or people or any living or non-living thing) and have a relation ρ between them. Then we write that X is related by a relation ρ , to Y. Or X ρ Y.
And if ρ is a symmetric relation, we might say that Y is (also) related by a relation ρ to X. Or Y ρ X.
So, in one line; $X \rho Y \iff Y \rho X$ is true.

A relation is reflexive if X is related to itself by a relation. i.e., $X \rho X$. Consider the statement “Jen, John and Nick live in a house altogether.” once again. Is the relation of living reflexive? How to check? Ask like, Jen lives with Jen, true? Yes! Jen lives there.
A relation is transitive, means that some objects X, Y, Z are such that if X is related to Y by the relation, Y is related to Z by the relation, then X is also related to Z by the same relation.
i.e., $X \rho Y \wedge Y \rho Z \Rightarrow X \rho Z$. For example, the relationship of brotherhood is transitive. (Why?) Now we are able to define the equivalence relation.
We say that a relation ρ is an equivalence relation if following properties are satisfied: (i) $X \rho Y \iff Y \rho X$
(ii) $X \rho X$
(iii) $X \rho Y \ Y \rho Z \Rightarrow X \rho Z$.

Functions: Let f be a relation (we are using f at the place of earlier used ρ ) on an ordered pair $(x,y) : x \in X \ y \in Y$. We can write xfy, a relation. This relation is called a function if and only if for every x, there is always a single value of y. I mean to say that if $xfy_1$ is true and $xfy_2$ is also true, then always $y_1=y_2$. This definition is standard but there are some drawbacks of this definition, which we shall discuss in the beginning of Real Analysis .
Many synonyms for the word ‘function’ are used at various stages of mathematics, e.g. Transformation, Map or Mapping, Operator, Correspondence. As already said, in ordered pair (x,y), x is called the element of domain of the function (and X the domain of the function) and y is called the element in range or co-domain of the function (and Y the range of the function).

Here I will stop myself. I don’t want a post to be long (specially when writing on basic mathematics) that reader feel it boring. The intermediate mathematics of functions is planned to be discussed in Calculus and advanced part in functional analysis. Please note that I am regularly revising older articles and trying to maintain the accuracy and completeness. If you feel that there is any fault or incompleteness in a post then please make a comment on respective post. If you are interested in writing a guest article on this blog, then kindly email me at mdnb[at]live[dot]in.

# Introduction

In English dictionary, the word Set has various meanings. It is often said to be the word with maximum meanings (synonyms). But out of all, we should consider only one meaning: ”collection of objects” — a phrase that provides you enough clarity about what Set is all about. But It is not the exact mathematical definition of Set . The theory of Set as a mathematical discipline rose up with George Cantor, German mathematician. It is said that Cantor was working on some problems in Trigonometric series and series of real numbers, which accidently led him to recognise the importance of some distinct collections and intervals. And he started developing Set Theory. Well, we are not here to discuss the history of sets; but Mathematical importance.

Cantor defined the set as a ‘plurality concieved 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, concieved as a whole. The objects (or things) are called the elements or members of the set $S$. Some sets which are often termed 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 concept with no material existence but ‘Bird’ or ‘birds’ are real.

# Representing sets

Sets are represented in two main ways:
1. Standard Method: In this method we use to write all elements of a set in a curly bracket ( { } ).
For example:
Flock of Birds := {Bird-1, Bird-2, …, Bird-100,…}
or, $F:= \{B_1, B_2, \ldots, B_{100}, \ldots \}$ is a set.
Here I have used first capital letter of each term to notate the example mathematically. We read this set as, A set F is defined by a collection of objects $B_1, B_2, \ldots$ etc.
2. Characteristic Method: In this method, we write a representative element and define that by a characteristic property. A characteristic property of a set is a property which is satisfied by each member of that set and by nothing else.
For example, above set of Flock of birds can also be written as:
$\mathrm{ F := \{ B : B \ is \ a \ bird \} }$
which has the same meaning at a wider extent. We read it as: ”A set F is defined by element B such that B is a bird.”

# Standard Sets

Some standard sets in Mathematics are:

Set of Natural Numbers: It includes of the numbers, which we can count, viz. $\mathrm{ \{0,1,2,3,4,5,6,7, \ldots \}}$. The set of natural numbers is denoted by $\mathbb{N}$.

Set of Integers: Integers includes of negatives of natural numbers and natural numbers itself. It is denoted by $\mathbb{Z}$. $-5, -4, 1, 2, 0$ …all are integers. The rigorous definition of integers be discussed in fourth part of the series.

Set of Rational Numbers: Rational numbers are numbers which might be represented as $\frac{p}{q}$, where p and q both are integers and relatively prime to each other and q not being zero. The set of rational numbers is represented by $\mathbb{Q}$ and may include elements like $\frac{2}{3}, \frac{-5}{7}, 8$. The characteristic notation of the set of rational numbers is $\mathbb{Q} := \{ \dfrac{p}{q};/ p,q \in \mathbb{Z}, \ (p,q) \equiv 1 \ q \ne 0 \}$. The rigorous dicussion about rational numbers will be provided in fourth part of the series.

Empty Set: It is possible to conceive a set with no elements at all. Such a set is variously known as an empty set or a void set or a vacuous set or a null set.
An example of emptyset is the set $\{\mathrm{x:\ x \ is \ an \ integer \ and \ x^2=2} \}$, since there exists no integer which square is 2 —the set is empty. The unique empty set is denoted by $\emptyset$.
Unit Set: A set with only one element is called the unit set. {x} is a unit set.

Universal Set: A set which contains every possible element in the universe, is a universal set. It is denoted by $U$.

# Two Sets

Let $A$ and $B$ be two sets. We say that $A$ is a subset of $B$ (or $B$ is superset of $A$ or $A$ is contained in $B$ or $B$ contains $A$) if every element of $A$ is also an element of set $B$. In this case we write, $A \subseteq B$ or $B \supseteq A$ respectively, having the same meaning .
Two sets are equal to each other if and only if each is a subset of the other. Subset word might be understood using ‘sub-collection’ or ‘subfamily’ as its synonyms.
Operations on Sets:
As Addition, Subtraction, Multiplication and Division are the most common mathematical operations between numbers; Union, Intersection, Complement, Symmetric difference, Cartesian Products are the same between sets.

UNION OF SETS

If A and B are two sets, then their union (or join) is the set, defined by another set $S \cup T$ such that it consists of elements from either A or B or both. If we write the sets A and B using Characteristic Method as,

$\mathrm{A:= \{x : x \ is \ an \ element \ of \ set \ A\}}$.
and,$\mathrm{B:=\{x : x \ is \ an \ element \ of \ set \ B\}}$
then the union set of A and B is defined by set J such that

$\mathrm{J := \{x: x \ is \ an \ element \ of \ set \ A \ or \ set \ B \ or \ both \}}$.

For practical example, let we have two sets:
$A:= \{1,2,3,r,t,y\}$ and $B:=\{3,6,9,r,y,g,k\}$ be any two sets; then their union is $A \cup B :=\{ 1,2,3,6,9,r,t,y,g,k \}$.
Note that it behaves like writting all the elements of each set, just caring that you are not allowed to write one element twice.

Here is a short video explaining Unions of Sets:

INTERSECTION OF SETS

Intersection or meet of two sets A and B is similarly defined by ‘and’ connective. The set {x: x is an element of A and x is an element of B} or briefly $\mathrm { \{ x: x \in A \wedge x \in B \}}$. It is denoted by $A \cap B$ or by $A \cdot B$ or by $AB$.
For example, and by definition, if A and B be two sets defined as,
$A:=\{1,2,3,r,t,y\}$
$B:=\{3,6,9,r,g,k\}$
then their intersection set, defined by $A \cap B:= \{3,r\}$.

In simple words, the set formed with all common elements of two or more sets is called the intersection set of those sets.
Here is a video explaining the intersection of sets:

If, again, A and B are two sets, we say that A is disjoint from B or B is disjoint from A or both A and B are mutually disjoint, if they have no common elements. Mathematically, two sets A and B are said to be disjoint iff $A \cap B := \emptyset$ .
If two sets are not disjoint, they are said to intersect each other.

PARTITION OF A SET

A partition set of a set X is a disjoint collection of non-empty and distinct subsets of X such that each member of X is a member of exactly one member (subset) of the collection.
For example, if $\{q,w,e,r,t,y,u\}$ is a set of keyboard letters, then $\{ \{q,w,e\}, \{r\}, \{t,y\},\{u\}\}$ is a partition of the set and each element of the set belongs to exactly one member (subset) of partition set. Note that there are many partition sets possible for a set. For example, $\{\{q,w\}, \{e,r\},\{t,y,u\}\}$ is also a partition set of set $\{q,w,e,r,t,y,u\}$.
A Video on Partition of set:

COMPLEMENT SET OF A SET

The complement set $A^c$ of a set $A$ is a collection of objects which do not belong to $A$. Mathematically, $A^c := \{x: x \notin A \}$.

The relative complement of set $A$ with respect to another set $X$ is $X \cap A^c$ ; i.e., intersection of set $X$ and the complement set of $A$. This is usually shortened by $X-A$, read X minus A. Thus, $X-A := {x : x \in X \wedge x \notin A}$, that is, the set of members of $X$ which are not members of $A$.

The complement set is considered as a relatative complement set with respect to (w.r.t) the universal set, and is called the Absolute Complement Set.

A Video on Complement  of  A Set:

SYMMETRIC DIFFERENCE

The symmetric difference is another difference of sets $A$ and $B$, symbolized $A \Delta B$, is defined by the union of mutual complements of sets $A$ and $B$, i.e., $A \Delta B := (A -B) \cup (B-A) = B \Delta A$.

# Theorems on Sets

1. $A \cup (B \cup C) = (A \cup B) \cup C$
2. $A \cap (B \cap C) = (A \cap B) \cap C$
3. $A \cup B= B \cup A$
4. $A \cap B= B \cap A$
5. $A \cup (B \cap C)= (A \cup B) \cap (A \cup C)$
6. $A \cap (B \cup C)= (A \cap B) \cup (A \cap C)$
7. $A \cup \emptyset= A$
8. $A \cap \emptyset= \emptyset$
9. $A \cup U=U$
10. $A \cap U=A$
11. $A \cup A^c=U$
12. $A \cap A^c=\emptyset$
13. If $\forall A \ , A \cup B=A$ $\Rightarrow B=\emptyset$
14. If $\forall A \ , A \cap B=A \Rightarrow B=U$
15. Self-dual Property: If $A \cup B =U$ and $A \cap B=\emptyset \ \Rightarrow B=A^c$
16. Self Dual: ${(A^c)}^c=A$
17. ${\emptyset}^c=U$
18. $U^c= \emptyset$
19. Idempotent Law: $A \cup A=A$
20. Idempotent Law: $A \cap A =A$
21. Absorption Law: $A \cup (A \cap B) =A$
22. Absorption Law: $A \cap (A \cup B) =A$
23. de Morgen Law: ${(A \cup B)}^c =A^c \cap B^c$
24. de Morgen Law: ${(A \cap B)}^c =A^c \cup B^c$

# Another Theorem

The following statements about set A and set B are equivalent to one another

1. $A \subseteq B$
2. $A \cap B=A$
3. $A \cup B =B$

I trust that we are familiar with the basic properties of complements, unions and intersections. We should now turn to another very important concept, that of a function. So how to define a function? Have we any hint that can lead us to define one of the most important terms in mathematics? We have notion of Sets. We will use it in an ordered manner, saying that an ordered pair.
First of all we need to explain the the notion of an ordered pair. If $x$ and $y$ are some objects, how should we define the ordered pair $(x,y)$ of those objects? By another set? Yes!! The ordered pair is also termed as an ordered set. We define ordered pair $(x,y)$ to be the set $\{\{x,y\},\{x\}\}$. We can denote the ordered pair $(x,y)$ by too, if there is a desperate need to use the small bracket ‘( )’ elsewhere.
So, note that Ordered Pairs
$(x,y) := \{\{x,y\},\{x\}\}$
and $(y,x) :=\{\{y,x\},\{y\}\} =\{\{x,y\},\{y\}\}$ are not identical. Both are different sets.
You might think that if ordered pair can be defined with two objects, then why not with three or more objects. As we defined ordered pair (ordered double, as a term) $(x,y)$, we can also define $(x,y,z)$, an ordered triple. And similarly an ordered $n$ -tuple $(x_1, x_2, \ldots x_n)$ in general such that:
$(x,y):=\{\{x,y\},\{x\}\}$
$(x,y,z):=\{\{x,y,z\},\{x,y\},\{x\}\}$
$(x_1, x_2, x_3, \ldots x_n) := \{\{x_1, x_2, x_3, \ldots x_n\}, \{x_1, x_2, x_3, \ldots x_{n-1}\}, \ldots, \{x_1, x_2\}, \{x_1\}\}$.
Another important topic, which is very important in process to define function (actually in process to define ordered pair) is Cartesian Product (say it, Product, simply) of two sets. Let $A$ and $B$ be two sets. Then their Product (I said, we’ll not use Cartesian anymore) is defined to be the (another) set of an ordered pair, $(a,b)$, where $a$ and $b$ are the elements of set $A$ and set $B$ respectively. Mathematically; the product of two sets $A$ and $B$
$A \times B := \{(a,b) : a \in A, \ b \in B\}$.
Note that $A \times B \ne B \times A$.
The name as well as the notation is suggestive in that if $A$ has $m$ elements, $B$ has $n$ elements then $A \times B$ indeed has $mn$ elements.
We see that if we product two sets, we get an ordered pair of two objects (now we’ll say them, variables). Similarly if we product more than two sets we get ordered pair of same number of variables. For example:
$X \times Y := (x,y); x \in X, y \in Y$.
$X \times Y \times Z :=(x,y,z); x \in X, y \in Y, z \in Z$. etc.
The sets, which are being product are called the factor sets of the ordered pair obtained. When we form products, it is not necessary that the factor sets be distinct. The product of the same set $A$ taken $n$ times is called the $n$ -th power of $A$ and is denoted by $A^n$. Thus, $A^2$ is $A \times A$. $A^3$ is $A \times A \times A$. And so on.

Now we are ready to define functions. The next part of this series will focus on functions.