How to interpret mathematical negation in terms of logic An example of negation which is meant specifically for the question.

Statement: (∀x∈N)(∃y∈N)(x+y = 1), Negative: (∃x∈N)(∀y∈N)(x+y≠1)

A real life implication

Statement: For every hour, there is a man who dies, Negative: There exists an hour in which all men don't die


How do we interpret the mentioned example in the means of logic? I thought that the following negation is enough to conclude the right negative meaning of the statement

There exists an hour in which a man doesn't die

So what's the point in terms of logic to negate

A man

to

All men

?
 A: Some man not dying is not the negation of some man dying, for both can be true at the same time. 
So rather: if it is not true that some man dies, then it is true for all men that they do not die.
A: The statement "For every hour, there is a man who dies" is saying $$\forall h\,\exists m\, D(m, h)$$ where $D(x,y)$ means man $x$ died in hour $y$. 
The negation of this statement is clearly $\exists h\,\forall m\, \lnot D(m,h)$. 
Translating back to English, we have: there is an hour in which no man dies, i.e. all men don't die.
A: So you want to negate the statement, "For every hour, there is a man who dies."
Let's see intuitively when this statement is true.  This statement should be true if, at any given hour, you can find a man who dies.
So, when is it false then?  Well, it's definitely false if you can find a single hour such that no man dies during that hour.  Right?  But if no man dies, that means every man lives.
So it's not enough to find an hour such that a single man lives.  There could still be another man somewhere else who has died.
A: 
Statement: For every hour, there is a man who dies. 

$$\forall x:[H(x) \implies \exists y:[M(y) \land D(y,x)]]$$
where


*

*$H(x)$ means $x$ is an hour in time

*$M(y)$ means $y$ is a man

*$D(y,x)$ means $y$ dies in $x$



Negative: There exists an hour in which all men don't die

$$\exists x:[H(x) \land \forall y:[M(y) \implies \neg D(y,x)]]$$
Hint: Makes use of quantifier switching, the definition of implication and the removing of double negations.
A: $∀x.P(x)=true$ means "for every $x$, $P(x)$ is true".
We know that it is negation is "There is at least one $x$ such that $P(x)$ is false". It is symbollically written as $∃x.P(x)=false$.
This is a similar idea with if you want support a statement, you prove it for all $x$, or if you disagree you prove it by giving a (at least one) counter-example.
So this is how ∀ and ∃ end up being inverse of each other.
