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# Monthly Archives: August 2011

## 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 that I should 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.
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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 $\Bar{A} = U-A = \{ x | x \notin A \}$ where $U$ is universal set.
• Relative Complement of a set $A \, \text{relative to another set} \, X$ is given by $X-A=X\cap \Bar{A}=\{ x \in X | x \notin A\}$.
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. $A \cup \Bar{A}=U$
12. $A \cap \Bar{A}=\emptyset$
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 $A \cap B=\emptyset \ \Rightarrow B=\Bar{A}$
16. Self Dual: $\Bar{\Bar{A}}=A$
17. $\overline{\emptyset}=U$
18. $\Bar{U}= \emptyset$
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: $\overline{A \cup B} =\Bar{A} \cap \Bar{B}$
24. de Morgen Law: $\overline{A \cap B} =\Bar{A} \cup \Bar{B}$
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 $g \subseteq f$.

## Just another way to Multiply

Multiplication is probably the most important elementary operation in mathematics; even more important than usual addition. Every math-guy has its own style of multiplying numbers. But have you ever tried multiplicating by this way?
Exercise:
$88 \times 45$=?
Ans: as usual :- 3960 but I got this using a particular way:
88            45
176          22
352           11
704            5
1408          2
2816          1

Sum of left column=3960

Thus, $88 \times 45=3960$ (as usual).
You might be thinking that what did I do here. Okay, let we understand this method by illustrating another multiplication, of 48 with 35.

Step 1. Write the numbers in two separate columns.

$48 \ 35$

Step 2. Now, double the number in left column and half the number in right column such that the number in right column reduces to 1. If the number [remaining] in right column is odd, then leave the fractional part and only write integer part.

$48 \ 35 \\ 96 \ 17\\192 \ 8\\384 \ 4\\ 768 \ 2 \\ 1536 \ 1$

Step 3: Cancel out any number in the left column whose corresponding number in the right column is even.

48                       35
96                       17
192                      8
384                       4
768                       2
1536                      1

Step 4:Sum all the numbers in the left column which are not cancelled. This sum is the required product.

$=1680$

I agree this method of multiplying numbers is not easy and you’re not going to use this in your every day math. It’s a bit boring and very long way of multiplication. But you can use this way to tease your friends, teach juniors and can write this into your own NOTEBOOK for future understandings. Remember, knowing more is getting more in mathematics. [LOL] I don’t know who, silly else me, made this quote. Have Fun.

## My Favorite Math Comics-1: How To Email Your Professor:

I read Math Comics regularly and This is the one of My Favorite Math comics.

Comics Via spikedmath.com

I read this comic on November 2010 and Since then It was saved to my Mobile. Just to share with you. Click On the image to go to the Main Resource page. Taken Via Creative Commons License 2.5

## Nanostory of Nanotechnology

Fullerene

Well, this is not going to be a nano [very short] story either of fairies or aliens. This is a big story of Nanotechnology, one of the most advanced topics in physics. Wait. It’s not going to be so hard or advanced to read. It is really going to be a good story because I’m not going to teach you about this stuff. I am trying to say and save it’s history on MY DIGITAL NOTEBOOK. I think you all should also read this.
Nanotechnology has become a widely discussed topic today in newspapers, magazines, journals, blogs and even in television ads. It’s very common that some organization announcing ‘yet another’ “nano-conference”. Nanotechnology or nano tech in short, refers to the technology of creating materials, devices and functions using atomically manipulated matter. If you didn’t understand it clearly yet read further lines. (more…)