logic: two simple math contradictions 1.The contradiction of the sentence:  
- There is a greater number than a million.
can be stated as follows:
- There is a number which is not greater than a million.
2.and the contradiction of the sentence:
- For all x, there exist y such that it's smaller then it
can be stated as follows:
- There exist x such that, no number y is not smaller then it
At first I thought that #1 is wrong, that the proper contradiction will be
-There is not a greater number then a million.
but then I thought it could also be 
-All numbers are small or equal to a million
which makes
-There exist a number which is not greater then a million.
please help me with my thought process and the questions
with #2 I'm really clueless, at first I thought it true (it's the proper contradiciton).
but then I thought it should be
-There exist x such that, ALL numbers y are not smaller then it
to be proper contradiction.
 A: Neither (1) nor (2) is correct.
Let us first clarify some concepts - what you seem to be getting at is called the negation of a statement. It makes no sense to talk about the "contradiction" of a given statement. Instead, we say that two or more statements contradict each other, are in contradiction, or give rise to a contradiction if they cannot simultaneously be true.

(1)
The negation of 


*

*"There is a number greater than a million." 


is the statement:


*

*"It is is not true that there is a number greater than a million" $\Leftrightarrow$

*"There is not a number greater than a million." $\Leftrightarrow$

*"Every number is less than or equal to a million",  
which neither implies nor is implied by 


*

*"There exists a number which is not greater than a million." 


(If you also assume that there exists some number, then the former statements imply the latter. The latter do not imply the former, as there could exist two numbers - one less than a million, and one greater than a million.) 

(2)
Assuming the second statement is supposed to read 


*

*"For all $x$, there exists $y$ such that $y$ is smaller than $x$", 


then its negation is 


*

*"There exists an $x$ such that, for all $y$, $y$ is not smaller than $x$" $\Leftrightarrow$

*"There exists an $x$ such that there is no $y$ smaller than it", 
which is not the same as the statement that you suggested:


*

*"There exists an $x$ such that there is no $y$ not smaller than it" $\Leftrightarrow$

*"There exists an $x$ such that every $y$ is smaller than $x$".


Indeed, in the former case you are saying that $x$ is a least element, whereas in the latter case you are saying that $x$ is a greatest element. The original (unnegated) statement was saying that there is no least element.

I haven't really gone into the mechanics of negating statements, but in my opinion this is one of those times where studying concrete examples is more helpful than discussing the rules of negation at an abstract level.
A: You are asking about negation, not contradiction.   For the first, negating "There is a number greater than a million", you line "At first I thought" is correct.  The first proposed negation "There is a number that is not greater than a million" fails two ways.  In the natural numbers, there is a number greater than a million and there is also a number not greater than a million.  
The general principle is that to negate a statement "There exists x such that P(x)" you want to say "There does not exist x such that P(x)".  The first suggestion says "There exists x such that not P(x)" but the initial statement doesn't care if there are x's such that not P(x) (like numbers less than a million).  From "There does not exist x such that P(x)" you can then go to "For all x, not P(x)"
The second has two levels of quantifier, so you need to take the negation inside in stages.  The general rule is to negate a quantified sentence, you change the quantifier and negate what is inside.  So to negate "For all x there exists y such that $ y\lt x$" you say:
Not(For all x there exists y such that $ y\lt x$)
There exists x such that not(there exists y such that $ y\lt x$)
There exists x such that for all y not($ y\lt x$)
There exists x such that for all y $ x \le y$
A: Basically, you wer on the right track; the only suggestion is, write down the formulas step-by-step.
(i) The contradiction is simply negation : so we must start from the negation of $P$, i.e. $\lnot P$ :

$\lnot \exists n (n > Million)$.

(ii) Then we apply the rule about the "interdefinibility" of quantifiers : $\exists$ is $\lnot \forall \lnot$, so that : $\lnot \exists$ is $\forall \lnot$, and we get :

$\forall n \lnot (n > Million)$.

(iii) Finally, we negate the condition inside the parentheses :

$\forall n (n \le Million)$.

About the second formula, proceed in the same way, applying the rules of the quantifiers twice :

$\lnot \forall x \exists y (y < x)$

i.e.

$\exists x \lnot \exists y (y < x)$

i.e.

$\exists x \forall y \lnot (y < x)$

i.e.

$\exists x \forall y (y \ge x)$

A: Where P(x) is the statement $x>1000000$
$\neg\exists x\in X\space P(x)=\forall x\in X\space \neg P(x)$
So basically it's
Every number is smaller then $1000000$
The second one is mathematically correct(though it should be There exist x such that, no number y is smaller then it but I guess your language uses double negation)
