Could someone explain DeMorgan Laws? I'm having a bit of trouble visualizing these laws we learned in class today.  He mentioned DeMorgan's Law when dealing with Quantifiers, and wrote this on the board:
$$\neg \forall x P(x) \iff \exists x \neg P(x)$$
$$\neg \exists x P(x) \iff \forall x \neg P(x)$$
I can't really visualize the two in my head (at least with real world examples), would someone be able to spell it out for me? (I'm also not familiar with MathJax or any of that so I pretty much copied/pasted it from my online notes, I hope it goes through ok)
 A: To understand this kind of thing, I recommend simple real world examples.  For the first one, let's take the universe to be the set of all birds, and $P(x)$ to mean "$x$ can fly".  Then


*

*$\forall x\,P(x)$ means "all birds can fly";

*$\neg\forall x\,P(x)$ means "not all birds can fly". . . 

*. . . and this is the same as "there are birds which cannot fly". . .

*. . . which can be written $\exists x\,\neg P(x)$.


Other equivalences can be interpreted similarly: you will probably learn the subject best if you try to make up your own examples as far as possible.

BTW I have never actually heard these referred to as "De Morgan's Laws".  I believe that term is usually taken for the equivalence
$$\neg(p\vee q)\quad\Leftrightarrow\quad (\neg p)\wedge (\neg q)$$
and its dual; though it is true that these are related to the ones you have quoted.
A: Here is an example to make it clearer.
Use de morgan's law to write the negation of $x$ more than $-1$ and less than $4$.
The given statement is equivalent to $x>-1$ AND $x<4$
By de morgan's law , the negation is
$x$ must be less than or equals to $-1$
               OR
$x$ must be more than or equals to $4$.
Pictorially you will understand it if you draw a line and label the numbers.
