SA,

it's my first time to write a technical topic on the blog, as i wanted this blog to be my personal not technical pages .. discuss my life and my opinions
But ....
5 days ago. in my ACM training Eng. M.M.A.Wahab told me that:
"There are the same number of rational numbers as natural numbers" (how .. rational are more??)
"There are the same number of Integer numbers as natural numbers" (Are you kidding me ?)

i didn't believe he tried to convene me, But i didn't...
After i went back to home, i started to Google that.. and i opened the discrete mathematics book to check that.

this is like a ZIGZAG shape i have started from
0 associate it with 0
then
1/2 associate it with 1
then
-1/1 associate it with 2
... etc.(neglect repeated values)

using this method we can give every rational number a natural number..
and the correspondence will be like that :

and that matching will cover all of Q and all of natural numbers.

Is this relation Injective ??
yes it's injective as for every number in natural numbers there exists only one value of rational number it could match
Is this relation Surjective ??
yes, as there is no value in the co-domain rational number that can't be mapped to one natural number

So this relation is bijective --> then the cardinality of the domain = the cardinality of co-domain
|Q| = |N| #R.T.P

-----------------------------------

2- "There are the same number of Integer numbers as natural numbers"

Let us define functions :

a- abs(X) = X if(X>=0)
abs(X) = -1*X if(X<0)

b- sign(X) = 1 if(x>0)
sign(X) = 0 if(X<=0)

F(X) --> 2*abs(X) - sign(X)

where X belongs to Integers and F(X) belongs to natural numbers
F(0) = 0, F(1) = 1, F(-1) = 2, F(2) = 3 ....

• This function is Injective where every element in Co-domain(Natural Numbers) there exists only 1 Integer number it could match

• This function is Surjective where every element in Co-domain is covered using a corresponding value in domain

So this function is bijection "Cardinality of Co-domain = Cardinality of Domain"
then
|Integer Numbers| = |Natural Numbers| ..... #R.T.P

DONE :D

what a mess ?? How can the cardinality of Integer Numbers = Cardinality of Natural Numbers
are Negative Numbers don't exist :S

But ... these proofs are right and you can find it on discrete and mathematics book when you search about countable sets & Contar's thearom

But i think that's all because INFINITY is not defined.. if infinity was defined all these theorem and proofs will be wrong...

BUT FOR NOW THEY ARE TRUE

Hello,
long time no see,
you know it's midyear vacation.

PS: "it's called midyear vacation, But it wasn't my midyear vacation as i had many many many things to do and I'm going to talk about that on a later post isAllah"

so, lets start this topic :

A week ago, i was at my coach's home, we were talking about TopCoder forum topic called : "you know you are a TopCoder when .."

He actually was talking about a situation happens to us all (C++,JAVA programmers):
when we are chatting or writing statements any where "we terminate our statements with semicolon instead of dot"

The important things is, when i went home this night i open the TopCoder forum and start looking for this topic, then i found it start reading it.
i liked this topic very much it's amazing really, this topic had started from 2003 and still active till now.

I book marked it and every day i check it and read few pages.
here are some quotes i find in this topic and liked them very much:


You Know You are a TopCoder when :

... When you see a sunrise and think "a flood-fill algorithm would have done that quicker";
... When you include your TopCoder rating changes in your weekly progress reports to you manager;
... When you think red > yellow > blue > green > gray, and orange is a special color;
... When you believe that "TopCoder - The Movie" would be a major box office success;
... When you start signing your emails with TopCoder Handle only;
... After paste your info into any web form you look for button "Compile";
... Instead of asking for directions, you just DFS;
... When you go out the balcony and see only red in front of your eyes;
... When part of your "life plan" is to improve your "color";
... When you are constantly removing the final t when typing in;
... When you carry a TopCoder code template in your wallet;
... When your friends call you by your TopCoder handle and not by your name;
... When the first thing your mother asks on a visit is if your rating is okay;
... When you declare variables during your math classes;
... When you believe "all time" started with SRM1;