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.
It was a big shock to find out that is true... i found theorems and proofs about that..
so i managed to proof that here.
so i managed to proof that here.
--------------------------------------------------------------------------
Before starting my proofs i have to define some terminologies:1- Function:
In mathematics F(X) said to be a function where for every X in it's domain it produce a value on corresponding co-domain
2- Injective function:
In mathematics injective function is a function which associates distinct arguments with distinct values; that is, every unique argument produces a unique result (wikipedia)
3- Surjective function:
In mathematics, a function f is said to be surjective, if its values span its whole co-domain; that is, for every y in the co-domain there is at least one x in the domain such that f(x) = y (wikipedia)
4- Bijective function:
In mathematics, a bijective function is a function f from a set X to a set Y with the property that, for every y in Y there is exactly one x in X such that f(x) = y.
Alternatively, f is bijective if it is injective & surjective (wikipedia)
if f(x)->y is bijective then the cardinality of the domain X = the cardinality of the co-domain Y
-----------------------------------------------------------------------------------
So lets move to our proofs.1- "There are the same number of rational numbers as natural numbers"
Assume Q is rational numbers, N is natural number
we must exhibit a one-to-one relationship"bijection" between Q & N
we will write Q as :
1st row denominator 1, 2nd row denominator 2 .. etc.
we can see that as a rectangular array that contain all rational numbers .. But there are some repetitions like 1/1 & 2/2 ....
Now we will try to map every rational number to a natural number (counting)
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
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...
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 :
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)
So this function is bijection "Cardinality of Co-domain = Cardinality of Domain"
then
|Integer Numbers| = |Natural Numbers| ..... #R.T.P
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 ....
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
3 Comments:
Subscribe to:
Post Comments (Atom)
" the rational numbers = natural numbers "
&
" integer numbers = natural "
kdaa kteeeeeeeeer awyy wallahy :S:S
bas it was a good new piece of information, menkom nastafeed :)