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A Trip to Mathematics: Part V Equations-I

Saturday, December 3rd, 2011 14:25 / Leave a Comment

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 will we achieve it mathematically?’ —We shall restate the above problem as another statement (meaning the same):

George had ~~some money~~ $x$ dollars. He gave 14 dollars to Matthew. Now he has 27 dollars. ~~How much money he had?~~ Find the value of $x$ .

This is equivalent to the problem asked above. I have just replaced ‘some money’ by ‘x dollars’. As ‘some’ senses as unknown quantity— $x$ does the same. Now all we need to get the value of x.
When solving for $x$ , we should have a plan like this:

George had	$x$ dollars.
He gave to Matthew	14 dollars
Now he must have	$x-14$ dollars

But problem says that he has 27 dollars left. This implies that $x-14$ dollars are equal to 27 dollars.
i.e., $x-14=27$

$x-14=27$ contains an alphabet x which we assumed to be unknown—can have any certain value. Statements (like $x-14=27$ ) containing unknown quantities and an equality are called Equations. The unknown quantities used in equations are called variables, usually represented by bottom letters in english alphabet (e.g., $x,y,z$ ). Top letters of alphabet ( $a,b,c,d$ ..) are usually used to represent constants (one whose value is known, but not shown).

Now let we concentrate on the problem again. We have the equation x-14=27.
Now adding 14 to both sides of the equal sign:
$x-14 +14 =27 +14$
or, $x-0 = 41$ (-14+14=0)
or, $x= 41$ .
So, $x$ is 41. This means George had 41 dollars. And this answer is equal to the answer we found practically. Solving problems practically are not always possible, specially when complicated problems encountered —we use theory of equations. To solve equations, you need to know only four basic operations viz., Addition, Subtraction, Multiplication and Division; and also about the properties of equality sign.
We could also deal above problem as this way:
$x-14= 27$
or, $x= 27+14 =41$
-14 transfers to another side, which makes the change in sign of the value, i.e., +14.

When we transport a number from left side to right of the equal sign, the sign of the number changes and vice-versa. As here -14 converts into +14; +18 converts into -18 in example below:
$x+18 =32$
or, $x=32 -18 =14$ .
Please note, any number not having a sign before its value is deemed to be positive—e.g., 179 and +179 are the same, in theory of equations.
Before we proceed, why not take another example?

Marry had seven sheep. Marry’s uncle gifted her some more sheep. She has eighteen sheep now. How many sheep did her uncle gift?

First of all, how would you state it as an equation?
$7 + x = 18$
or, $+7 +x =18$ (just to illustrate that 7=+7)
or, $x= 18-7 =9$ .
So, Marry’s uncle gifted her 9 sheep. ///
Now tackle this problem,

Monty had some cricket balls. Graham had double number of balls as compared to Monty. Adam had also 6 cricket balls. They all collected their balls and found that total number of cricket balls was 27. How many balls had Monty and Graham?

As usual our first step to solve this problem must be to restate it as an equation. We do it like this:
Monty had (let) x balls.
Then Graham must had $x \times 2=2x$ balls.
Adam had 6 balls.
The total sum= $x+2x+6=3x+6$
But that is 27 according to our question.
Hence, $3x+6=27$
or, $3x=27-6 =21$
or, $x=21 /3 =7$ .
Here multiplication sign converts into division sign, when transferred.
Since $x=7$ , we can say that Monty had 7 balls (instead of x balls) and Graham had 14 (instead of $2x$ ).
///

Types of Equations

They are many types of algebraic equations (we suffix ‘Algebraic’ because it includes variables which are part of algebra) depending on their properties. In common we classify them into two main parts:

1. Equations with one variable (univariable algebraic equations, or just Univariables)

2. Equations with more than one variables (multivariable algebraic equations, or just Multivariables)

Univariable Equations

Equations consisting of only one variable are called univariable equations.

All of the equations we solved above are univariables since they contain only one variable (x). Other examples are:
$3x+2=5x-3$ ;
$x^2+5x +3=0$ ;
$e^x =x^e$ (e is a constant).

Univariables are further divided into many categories depending upon the degree of the variable. Some most common are:

Linear Univariables: Equations having the maximum power (degree) of the variable 1.
$ax+b=c$ is a general example of linear equations in one variable, where a, b and c are arbitrary constants.
Quadratic Equations: Also known as Square Equations, are ones in which the maximum power of the variable is 2.
$ax^2+bx+c=0$ is a general example of quadratic equations, where a,b,c are constants.
Cubic Equations: Equations of third degree (maximum power=3) are called Cubic.
A cubic equation is of type $ax^3+bx^2+cx+d=0$ ; where a,b,c,d are constants.
Quartic Equations: Equations of fourth degree are Quartic.
A quartic equation is of type $ax^4+bx^3+cx^2+dx+e=0$ .

Similarly, equation of an n-th degree can be defined if the variable of the equation has maximum power n.

Multivariable Equations:

Some equations have more than one variables, as $ax^2+2hxy+by^2=0$ etc. Such equations are termed as Multivariable Equations. Depending on the number of variables present in the equations, multivariable equations can be classified as:

1. Bi-variable Equations - Equations having exactly two variables are called bi-variables.
$x+y=5$ ; $x^2+y^2=4$ ; $r^2+\theta^2=k^2$ , where k is constant; etc are equations with two variables.
Bivariable equations can also be divided into many categories, as same as univariables were.

A.Linear Bivariable Equations: Power of a variable or sum of powers of product of two variables does not exceed 1.
For example: $ax+by=c$ is a linear but $axy=b$ is not.
B. Second Order Bivariable Equations: Power of a variable or sum of powers of product of two variables does not exceed 2.
For example: $axy=b$ , $ax^2+by^2+cxy+dx+ey+f=0$ are of second order.
Similarly you can easily define n-th order Bivariable equations.

2. Tri-variable Equations: Equations having exactly three variables are called tri-variable equations.
$x+y+z=5$ ; $x^2+y^2-z^2=4$ ; $r^3+\theta^3+\phi^3=k^3$ , where k is constant; etc are trivariables. (Further classification of Trivariables are not necessary, but I hope that you can divide them into more categories as we did above.)
Similary, you can easily define any n-variable equation as an equation in which the number of variables is n.

Out of these equations, we shall discuss only Linear Univariable Equations here (actually we are discussing them). ////

We have already discussed them above, for particular example. Here we’ll discuss them for general cases.
As told earlier, a general example of linear univariable equation is $ax+b=c$ .
We can adjust it by transfering constants to one side and keeping variable to other.
$ax+b = c$
or, $ax = c-b$
or, $x = \frac{c-b}{a}$
this is the required solution.
Example: Solve $3x+5=0$ .
We have, $3x+5=0$
or, $3x = 0-5 =-5$
or, $x = \frac{-5}{3}$ ////

A Trip to Mathematics: Part IV Numbers

Monday, November 21st, 2011 16:07 / Leave a Comment

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.”

What does it mean? What did Kronecker think when he made this quote? Why is this quote true? —First part of this article is based on this discussion.
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.

A Trip To Mathematics: Part II Set(Basics) (wpgaurav.wordpress.com)
Do complex numbers really exist? (math.stackexchange.com)
Definitions (gowers.wordpress.com)

A Trip to Mathematics: Part III Relations and Functions

Thursday, November 10th, 2011 11:37 / Leave a Comment

Read these statements carefully:
‘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.

Equivalence relations (gowers.wordpress.com)

A Trip To Mathematics: Part II Set(Basics)

Friday, November 4th, 2011 05:55 / 3 Comments

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

$A \cup (B \cup C) = (A \cup B) \cup C$
$A \cap (B \cap C) = (A \cap B) \cap C$
$A \cup B= B \cup A$
$A \cap B= B \cap A$
$A \cup (B \cap C)= (A \cup B) \cap (A \cup C)$
$A \cap (B \cup C)= (A \cap B) \cup (A \cap C)$
$A \cup \emptyset= A$
$A \cap \emptyset= \emptyset$
$A \cup U=U$
$A \cap U=A$
$A \cup A^c=U$
$A \cap A^c=\emptyset$
If $\forall A \ , A \cup B=A$ $\Rightarrow B=\emptyset$
If $\forall A \ , A \cap B=A \Rightarrow B=U$
Self-dual Property: If $A \cup B =U$ and $A \cap B=\emptyset \ \Rightarrow B=A^c$
Self Dual: ${(A^c)}^c=A$
${\emptyset}^c=U$
$U^c= \emptyset$
Idempotent Law: $A \cup A=A$
Idempotent Law: $A \cap A =A$
Absorption Law: $A \cup (A \cap B) =A$
Absorption Law: $A \cap (A \cup B) =A$
de Morgen Law: ${(A \cup B)}^c =A^c \cap B^c$
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

$A \subseteq B$
$A \cap B=A$
$A \cup B =B$

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.

Keep Reading and Commenting.

A Trip to Mathematics: Part-I Logic

Tuesday, October 25th, 2011 13:34 / 5 Comments

About “A Trip To Mathematics”:

A Trip to Mathematics is an indefinitely long series, aimed on generally interested readers and other undergraduate students. This series will deal Basic Mathematics as well as Advanced Mathematics in very interactive manners. Each post of this series is kept small that reader be able to grasp concepts. Critics and suggestions are invited in form of comments.

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. Let we discuss about Sentence &Statements, then we shall proceed to further logic .

Sentence & Statements

A sentence is a collection of some words, those together having some sense.

For example:

Math is a tough subject.
English is not a tough subject.
Math and English both are tough subjects.
Either Math or English is tough subject.
If Math is a tough subject, then English is also a tough subject.
Math is a tough subject, if and only if English is a tough subject.

Just have a quick look on above collections of words. Those are sentences, since they have some meaning too. First sentence is called Prime Sentence, i.e., sentence which either contains no connectives or, by choice, is regarded as “indivisible”. The five words

not

and

or

if …. then

if and only if

or their combinations are called ‘connectives‘. The sentences (all but first) are called composite sentences, i.e., a declarative sentence (statement) in which one or more connectives appear. Remember that there is no difference between a sentence and statement in general logic. In this series, sentences and statements would have the same meaning.

Connectives

not: A sentence which is modified by the word “not” is called the negation of the original sentence.
For example: “English is not a tough subject” is the negation of “English is a tough subject“. Also, “3 is not a prime” is the negation of “3 is a prime“. Always note that negation doesn’t really mean the converse of a sentence. For example, you can not write “English is a simple subject” as the negation of “English is a tough subject“.
In mathematical writings, symbols are often used for conciseness. The negation of sentences/statements is expressed by putting a slash (/) over that symbol which incorporates the principal verb in the statement.
For example: The statement $x=y$ (read ‘x is equal to y’) is negated as $x \ne y$ (read ‘x is not equal to y‘). Similarly, $x \notin A$ (read ‘x does not belong to set A‘) is the negation of $x \in A$ (read ‘x belongs to set A‘).
Statements are sometimes represented by symbols like p, q, r, s etc. With this notation there is a symbol, $\not$ or ¬ (read as ‘not’) for negation. For example if ‘p’ stands for the statement “Terence Tao is a professor” then $\not p$ [or ¬p] is read as ‘not p’ and states for “Terence Tao is not a professor.” Sometimes ~p is also used for the negation of p.
and: The word “and” is used to join two sentences to form a composite sentence which is called the conjunction of the two sentences. For example, the sentence “I am writing, and my sister is reading” is the conjunction of the two sentences: “I am writing” and “My sister is reading“. In ordinary language (English), words like “but, while” are used as approximate synonyms for “and“, however in math, we shall ignore possible differences in shades of meaning which might accompany the use of one in the place of the other. This allows us to write “I am writing but my sister is reading” having the same mathematical meaning as above.
The standard notation for conjunction is $\wedge$ , read as ‘and‘. If p and q are statements then their conjunction is denoted by $p \wedge q$ and is read as ‘p and q’.
or: A sentence formed by connecting two sentences with the word “or” is called the disjunction of the two sentences. For example, “Justin Bieber is a celebrity, or Sachin Tendulkar is a footballer.” is a disjunction of “Justin Bieber is a celebrity” and “Sachin Tendulkar is a footballer“.
Sometimes we put the word ‘either‘ before the first statement to make the disjunction sound nice, but it is not necessary to do so, so far as a logician is concerned. The symbolic notation for disjunction is $\vee$ read ‘or’. If p and q are two statements, their disjunction is represented by $p \vee q$ and read as p or q.

if….then: From two sentences we may construct one of the from “If . . . . . then . . .“; which is called a conditional sentence. The sentence immediately following IF is the antecedent, and the sentence immediately following THEN is the consequent. For example, “If 5 <6, 6<7, then 5<7” is a conditional sentence whith “5<6, 6<7” as antecedent and “5<7” as consequent. If p and q are antecedent and consequent sentences respectively, then the conditional sentence can be written as:

“If p then q”.

This can be mathematically represented as $p \Rightarrow q$ and is read as “p implies q” and the statement sometimes is also called implication statement. Several other ways are available to paraphrase implication statements including:

If p then q
p implies q
q follows from p
q is a logical consequence of p
p (is true) only if q (is true)
p is a sufficient condition for q
q is a necessary condition for p

If and Only If: The phrase “if and only if” (abbreviated as ‘iff‘) is used to obtain a bi-conditional sentence. For example, “A triangle is called a right angled triangle, if and only if one of its angles is 90°.” This sentence can be understood in either ways: “A triangle is called a right angled triangle if one of its angles is 90°” and “One of angles of a triangle is 90° if the triangle is right angled triangle.” This means that first prime sentence implies second prime sentence and second prime sentence implies first one. (This is why ‘iff’ is sometimes called double-implication.)

Another example is “A glass is half filled iff that glass is half empty.“

If p and q are two statements, then we regard the biconditional statement as “p if and only if q” or “p iff q” and mathematically represent by ” $p\iff q$ “. $\iff$ represents double implication and read as ‘if and only if’.

In the statement $p \iff q$ , the implication $p \Rightarrow q$ is called direct implication and the implication $q \Rightarrow p$ is called the converse implication of the statement.

Other terms in logic:

Stronger and Weaker Statements: A statement p is stronger than a statement q (or that q is weaker than p) if the implication statement $p \Rightarrow q$ is true.
Strictly Stronger and Strictly Weaker Statements: The word ‘stronger‘(or weaker) does not necessarily mean ‘strictly stronger‘ (or strictly weaker).
For example, every statement is stronger than itself, since $p \Rightarrow p$ . The apparent paradox here is purely linguistical. If we want to avoid it, we should replace the word stronger by the phrase ‘stronger than‘ or ‘possibly as strong as‘.

If $p \Rightarrow q$ is true but its converse is false ( $q \not \Rightarrow p$ ), then we say that p is strictly stronger than q (or that q is strictly weaker than p). For example it is easy to say that a given quadrilateral is a rhombus that to say it is a parallelogram. Another understandable example is that ” If a blog is hosted on WordPress.com, it is powered with WordPress software.” is true but ” If a blog is powered with WordPress software , it is hosted on WordPress.com” is not true.

Logical Approach

What exactly is the difference between a mathematician, a physicist and a layman?
Let us suppose they all start measuring the angles of hundreds of triangles of various shapes, find the sum in each case and keep a record.

Suppose the layman finds that with one or two exceptions the sum in each case comes out to be 180 degrees. He will ignore the exceptions and state ‘The sum of the three angles in a triangle is 180 degrees.’

A physicist will be more cautious in dealing the exceptional cases. He will examine then more carefully. If he finds that the sum in them some where 179 degrees to 181 degrees, say, then if will attribute the deviation to experimental errors. He will state a law – ‘The sum of the three angles of any triangle is 180 degrees.’ He will then watch happily as the rest of the world puts his law to test and finds that it holds good in thousands of different cases, until somebody comes up with a triangle in which the law fails miserably. The physicist now has to withdraw his law altogether or else to replace it by some other law which holds good in all the cases tried. Even this new law may have to be modified at a later date. And this will continue without end.

A mathematician will be the fussiest of all. If there is even a single exception, he will refrain from saying anything. Even when millions of triangles are tried without a single exception, he will not state it as a theorem that the sum of the three angles in ‘any’ triangle is 180 degrees. The reason is that there are infinitely many different types of triangles. To generalise from a million to infinity is as baseless to a mathematician as to generalise from one to a million. He will at the most make a conjecture and say that there is a ‘strong evidence’ suggesting that the conjecture is true.

The approach taken by the layman or the physicist is known as the inductive approach whereas the mathematician’s approach is called the deductive approach.

Inductive Approach

In inductive approach, we make a few observations and generalise. Exceptions are generally not counted in inductive approach.

Deductive Approach

In this approach, we deduce from something which is already proven.

Axioms or Postulates

Sometimes, when deducting theorems or conclusion from another theorems, we reach at a stage where a certain statement cannot be proved from any ‘other’ proved statement and must be taken for granted to be true, then such a statement is called an axiom or a postulate.
Each branch of mathematics has its own populates or axioms. For example, the most fundamental axiom of geometry is that infinitely many lines can be drawn passing through a single point. The whole beautiful structure of geometry is based on five or six such axioms and every theorem in geometry can be ultimately Deducted from these axioms.

Argument, Premises and Conclusion

An argument is really speaking nothing more than an implication statement. Its hypothesis consists of the conjunction of several statements, called premises. In giving an argument, its premises are first listed (in any order), then connecting all, a conclusion is given. Example of an argument:
Premises:   $p_1$          Every man is mortal.
$p_2$                               Ram is a man.
———————————————————————————-
Conclusion:                   $q$ Ram is mortal.

Symbolically, let us denote the premises of an argument by $p_1, p_2, \ldots , p_n$ and its conclusion by $q$ . Then the argument is the statement $(p_1 \wedge p_2 \wedge \ldots \wedge p_n) \Rightarrow q$ . If this implication is true, the argument is valid otherwise it is invalid.

To be continued……

Basic logic – connectives – NOT (gowers.wordpress.com)
Welcome to the Cambridge Mathematical Tripos (gowers.wordpress.com)

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Tag Archives: A Trip To Mathematics

A Trip to Mathematics: Part V Equations-I

Types of Equations

A Trip to Mathematics: Part IV Numbers

A Trip to Mathematics: Part III Relations and Functions

A Trip To Mathematics: Part II Set(Basics)

Introduction

Representing sets

Standard Sets

Two Sets

Theorems on Sets

Another Theorem

A Trip to Mathematics: Part-I Logic

About “A Trip To Mathematics”:

What is Logic?

Sentence & Statements

Connectives

Other terms in logic:

Logical Approach

Inductive Approach

Deductive Approach

Axioms or Postulates

Argument, Premises and Conclusion

Follow “MY DIGITAL NOTEBOOK”

MY DIGITAL NOTEBOOK

Tag Archives: A Trip To Mathematics

A Trip to Mathematics: Part V Equations-I

Types of Equations

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A Trip to Mathematics: Part IV Numbers

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A Trip to Mathematics: Part III Relations and Functions

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A Trip To Mathematics: Part II Set(Basics)

Introduction

Representing sets

Standard Sets

Two Sets

Theorems on Sets

Another Theorem

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A Trip to Mathematics: Part-I Logic

About “A Trip To Mathematics”:

What is Logic?

Sentence & Statements

Connectives

Other terms in logic:

Logical Approach

Inductive Approach

Deductive Approach

Axioms or Postulates

Argument, Premises and Conclusion

Suggested Readings:

Better You Share!

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