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.’

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.

A Trip to Mathematics: Part-I Logic

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.

The problem of the Hundred Fowls

This is a popular Chinese problem, on Linear Diophantine equations, which in wording seems as a puzzle or riddle. However, when used algebraic notations, it looks obvious. The problems states :

Fermat Numbers

Fermat Number, a class of numbers, is an integer of the form $F_n=2^{2^n} +1 \ \ n \ge 0$ .

For example: Putting $n := 0,1,2 \ldots$ in $F_n=2^{2^n}$ we get $F_0=3$ , $F_1=5$ , $F_2=17$ , $F_3=257$ etc.

How to Draw the Famous Batman Curve

The ellipse $\displaystyle \left( \frac{x}{7} \right)^{2} + \left( \frac{y}{3} \right)^{2} – 1 = 0$ looks like this:

So the curve $\left( \frac{x}{7} \right)^{2}\sqrt{\frac{\left| \left| x \right|-3 \right|}{\left| x \right|-3}} + \left( \frac{y}{3} \right)^{2}\sqrt{\frac{\left| y+3\frac{\sqrt{33}}{7} \right|}{y+3\frac{\sqrt{33}}{7}}} – 1 = 0$ is the above ellipse, in the region where $|x|>3$ and $y > -3\sqrt{33}/7$ :

Pi (Π)

Pi, the mathematical constant, has a yet to determine value. As of now following sequence of digits has found to be most accurate for the approximation for $pi$ . :3.1415926535897932384626433832795028841971693993751058209 7494459230781640628620899862803482534211706798214808651 3282306647093844609550582231725359408128481117450284102 7019385211055596446229489549303819644288109756659334461 2847564823378678316527120190914564856692346034861045432 6648213393607260249141273724587006606315588174881520920 9628292540917153643678925903600113305305488204665213841 4695194151160943305727036575959195309218611738193261179 3105118548074462379962749567351885752724891227938183011 9491298336733624406566430860213949463952247371907021798 6094370277053921717629317675238467481846766940513200056 8127145263560827785771342757789609173637178721468440901 2249534301465495853710507922796892589235420199561121290 2196086403441815981362977477130996051870721134999999837 2978049951059731732816096318595024459455346908302642522 3082533446850352619311881710100031378387528865875332083 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Announcement

Hi all!
I know some friends, who don’t know what mathematics in real is, always blame me for the language of the blog. It is very complicated and detailed. I understand that it is. But MY DIGITAL NOTEBOOK is mainly prepared for my study and research on mathematical sciences. So, I don’t care about what people say (SAID) about the

A Torus

content and how many hits did my posts get. I feel happy in such a way that MY DIGITAL NOTEBOOK has satisfied me at its peak-est level. I would like to thank WordPress.com for their brilliant blogging tools and to my those friends, teachers and classmates who always encourage me about my passion. For me the most important thing is my study. More I learn, more I will go ahead. So, today (I mean tonight) I have decided to write some lecture-notes (say them study-notes, since I am not a lecturer) on MY DIGITAL NOTEBOOK. I have planned to write on Group Theory at first and then on Real Analysis. And this post is just to introduce you with some fundamental notations which will be used in those study-notes.

Notations

Conditionals and Operators
$r /; c$ : Relation $r$ holds under the condition $c$ .
$a=b$ : The expression $a$ is mathematically identical to $b$ .
$a \ne b$ : The expression a is mathematically different from $b$ .
$x > y$ : The quantity $x$ is greater than quantity $y$ .
$x \ge y$ : The quantity $x$ is greater than or equal to the quantity $y$ .
$x < y$ : The quantity $x$ is less than quantity $y$ .
$x \le y$ : The quantity $x$ is less than or equal to quantity $y$ .
$P := Q$ : Statement $P$ defines statement $Q$ .
$a \wedge b$ : a and b.
$a \vee b$ : a or b.
$\forall a$ : for all $a$ .
$\exists$ : [there] exists.
$\iff$ : If and only if.
Sets & Domains
$\{ a_1, a_2, \ldots, a_n \}$ : A finite set with some elements $a_1, a_2, \ldots, a_n$ .
$\{ a_1, a_2, \ldots, a_n \ldots \}$ : An infinite set with elements $a_1, a_2, \ldots$
$\mathrm{\{ listElement /; domainSpecification\}}$ : A sequence of elements listElement with some domainSpecifications in the set. For example, $\{ x : x=\frac{p}{q} /; p \in \mathbb{Z}, q \in \mathbb{N^+}\}$ $a \in A$ : $a$ is an element of the set A.
$a \notin A$ : a is not an element of the set A.
$x \in (a,b)$ : The number x lies within the specified interval $(a,b)$ .
$x \notin (a,b)$ : The number x does not belong to the specified interval $(a,b)$ . Standard Set Notations
$\mathbb{N}$ : the set of natural numbers $\{0, 1, 2, \ldots \}$
$\mathbb{N}^+$ : The set of positive natural numbers: $\{1, 2, 3, \ldots \}$
$\mathbb{Z}$ : The set of integers $\{ 0, \pm 1, \pm 2, \ldots\}$
$\mathbb{Q}$ : The set of rational numbers
$\mathbb{R}$ : The set of real numbers
$\mathbb{C}$ : The set of complex numbers
$\mathbb{P}$ : The set of prime numbers.
$\{ \}$ : The empty set.
$\{ A \otimes B \}$ : The ordered set of sets $A$ and $B$ .
$n!$ : Factorial of n: $n!=1\cdot 2 \cdot 3 \ldots (n-1) n /; n \in \mathbb{N}$

Other mathematical notations, constants and terms will be introduced as their need.

For Non-Mathematicians:
Don’t worry I have planned to post more fun. Let’s see how the time proceeds!

381654729 : An Interesting Number Happened To Me Today

Image via Wikipedia

You might be thinking why am I writing about an individual number? Actually, in previous year annual exams, my registration number was 381654729. Which is just an ‘ordinary’ 9-digit long number. I never cared about it- and forgot it after exam results were announced. But today morning, when I opened “Mathematics Today” magazine’s October 2010, page 8; I was brilliantly shocked. 381654729 is a nine digit number with each of the digits from 1 to 9 appearing once. The whole number is divisible by 9. If you remove the right-most digit, the remaining eight-digit number is divisible by 8. Again removing the next-right-most digit leaves a seven-digit number that is divisible by 7. Similarly, removing next-rightmost digit leaves a six-digit number that is divisible by 6. This property continues all the way down to one digit.
Further research on this number provided a term for this number as Poly-divisible Number.
And I also noticed that a similar problem has been asked in U S A Mathematical Talent Search  competition.

After this beautiful incident, I would like to quote a statement here:

Mathematical Wonders happen with Mathematicians.

Numbers always chase me.

Four Math Problems

1. Solve the equation
${(\dfrac{1}{10})}^{\log_{\frac{x}{4}} {\sqrt [4] {x} -1}} -4^{\log_{10} {\sqrt [4] {x} +5}} =6, \forall x \ge 1$
2. The function $f: \mathbb{R} \to \mathbb{R}$ has $n+1$ derivatives. Show that if $a < b$ and $\log [f(b)+f'(b)+f"(b)+ \ldots +f^n(b)] – \log [f(a)+f'(a)+f"(a)+ \ldots +f^n(a)] =b-a$ then we can find $c \in (a,b)$ such that $f^{n+1} (c) = f (c)$ .
3. Let $A = {(x,y) : 0 \le < 1 }$ .
For $(x,y) \in A$ , let $\mathbf{S} (x,y)= \displaystyle{\sum_{\frac{1}{2} \le \frac{m}{n} \le 2}} x^m y^n$ where the sum ranges over all pairs $(m,n)$ of positive integer satisfying the indicated inequalities. Evaluate:
$\displaystyle {\lim_{{(x,y) \to (1,1)}_{(x,y) \in A}}} (1-xy^2)(1-x^2y) \mathbf{S} (x,y)$ .
4. This problem deals to elementary functional analysis and is taken from very old paper of Putnam Competitions.

$f: [0, \mathbf{N}] \to \mathbf{R}&s=1$ has continuous second derivative and $|f’(x)| < 1$ , $f”(x) > 0$ for all $x$ .
$0 \le m_0 < m_1 < m_2 < \ldots < m_k \le \mathbf{N}$ are integers such that $f(m_i)$ are all integers. Put $a_i=m_i-m_{i-1}$ and $b_i=f(m_i)-f(m_{i-1})$ .
•Prove that
$-1 < \frac {b_1}{a_1} < \frac {b_2}{a_2} < \ldots < < \frac {b_k}{a_k} < 1$ .
• Show that for $A > 1$ , there are at-most $\dfrac{\mathbf{N}}{A}$ such indices $i$ .
• Show that there are at most $3 {(\mathbf{N})}^{2/3}$ lattice points on the curve $y=f(x)$ .

Do you multiply this way!

Before my college days I used to multiply orthodoxly by this way.

But as time passed, I learned new things. I remember, In a Hindi magazine named “Bhaskar Lakshya”, I read an article in which a lecturer (apology, I don’t remember his name) had suggested how to multiply in single line (row). Today I thought to share this method on MY DIGITAL NOTEBOOK too.
I know there are many, who already know this method, but I think maximum people wouldn’t have any idea about this method. I found multiplicating this way, very faster – easier and smarter. The ‘only’ requirements for using this method is quick summation. You should be good in calculation and addition. Smarter your calculations, faster you’re.
I’ll try to illustrate this method below. If you had any problems regarding language (poor off-course) and understandings, please feel free to put that into comments.

1. Cantor’s Concept of a set

A set $S$ is any collection of definite, distinguishable objects of our intuition or of our intellect to be conceived as a whole. The objects are called the elements or members of set $S$

2. The intuitive principle of extension for sets

Two sets are equal if and only if (iff) they have the same members. i.e., $X=Y \, \Leftrightarrow \,\forall x \in X \text{and} \ x \in Y$ .

3. The intuitive principle of abstraction

A formula (syn: property) $P(x)$ defines a set $A$ by the convention that the members of $A$ are exactly those objects $a$ such that $P(a)$ is a true statement. $\Rightarrow a \in \{ x|P(x) \}$ .

4. Operations with/for sets

• Union (Sum or Join)$A \cup B= \{ x | x \in A \, \text{or} \, x \in B \}$
• Intersection (Product or Meet)
$A \cap B= \{ x| x\in A \, \text{and} \, x\in B \}$
• Disjoint Sets $A$ and $B$ are disjoint sets iff $A \cap B=\emptyset : \text{an empty set}$ and they intersect iff $A \cap B \ne \emptyset$
• Partition of Sets A partition of a set $X$ is a disjoint collection
$\mathfrak{X}$ of non-empty and distinct subsets of $X$ such that each member of $X$ is a member of some (and hence exactly one) member of $\mathfrak{X}$ .
For example: $\{ \{a,b\} \, \{c \} \, \{d, e\} \}$ is a partition of $\{a,b,c,d,e\}$ .
• Absolute Complement of a set $A$ is usually represented by $\overline{A} = U-A = \{ x | x \notin A \}$ where $U$ is universal set.
• Relative Complement of a set $A \, \text{relative to another set} \, X$ is given by $X-A=X\cap \overline{A}=\{ x \in X | x \notin A\}$ .
5. 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. $A \cup \overline{A}=U$
12. $A \cap \overline{A}=\emptyset$
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 $A \cap B=\emptyset \ \Rightarrow B=\overline{A}$
16. Self Dual: $\overline{\overline{A}}=A$
17. $\overline{\emptyset}=U$
18. $\overline{U}= \emptyset$
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: $\overline{A \cup B} =\overline{A} \cap \overline{B}$
24. de Morgen Law: $\overline{A \cap B} =\overline{A} \cup \overline{B}$
6. 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. $A \cup B =B$
7. Functions

Function is a relation such that no two distinct members have the same first co-ordinate in its graph. $f$ is a function iff

1. The members of $f$ are ordered pairs.
2. If ordered pairs $(x, y)$ and $(x, z)$ are members of $f$ , then $y=z$
8. Other words used as synonyms for the word ‘function’ are ‘transformation’, ‘map’, ‘mapping’, ‘correspondence’ and ‘operator’.
9. Notations for functions

A function is usually defined as ordered-pairs, see above, and $\text{ordered pair } (x,y) \in \text{function } f$ so that $xfy$ is (was) a way to represent where $x$ is an argument of $f$ and $y$ is image (value) of $f$ .
Other popular notations for $(x,y)\in f$ are: $y : xf$ , $y=f(x)$ , $y=fx$ , $y=x^f$ .

10. Intuitive law of extension for Functions

Two sets $f$ and $g$ are equal iff they have the same members (here, Domain and Range) $\Rightarrow f=g \Leftrightarrow D_f=D_g \ \text{and } \ f(x)=g(x)$

11. Into Function

A function $f$ is into $Y$ iff the range of $f$ is a subset of $Y$ . i.e., $R_f \subset Y$

12. Onto Function

A function $f$ is onto $Y$ iff the range of $f$ is $Y$ . i.e., $R_f=Y$

13. Generally a mapping is represented by $f : X \rightarrow Y$ .
14. One-to-One function

A function is called one-to-one if it maps distinct elements onto distinct elements.
A function $f$ is one-to-one iff $x_1 \ne x_2 \Leftrightarrow f(x_1) \ne f(x_2)$ and $x_1 = x_2 \Leftrightarrow f(x_1)=f(x_2)$

15. Restriction of Function

If $f : X \rightarrow Y$ and if $A \subseteq X$ , then $f \cap (A \times Y)$ is a function on $A \ \text{into } \ Y$ , called the restriction of $f$ to $A$ and $f \cap (A \times Y)$ is usually abbrevated by $f|A$ .

16. Extension of function

The function $f$ is an extension of a function $g$ iff $g \subseteq f$ .

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