Clarification needed regarding DeMorgan's Law I'm currently going through Daniel Velleman's "How to Prove It". He states DeMorgan's Law as follows:

1. $\neg (P\wedge Q)$ is equivalent to $\neg P \vee \neg Q$ 
2. $\neg (P \vee Q)$ is equivalent to $\neg P \wedge \neg Q$

The best way I knew to convince myself of the truth of these laws is by creating basic analogies. I used the following statements:

$P=$ I will eat strawberry ice cream 
  $Q=$ I will eat chocolate ice cream


1. $\neg (P \wedge Q)$ is equivalent to $\neg P \vee \neg Q$
$\neg (P \wedge Q)$: is equivalent to saying I will not eat both strawberry and chocolate ice cream. 
$\neg P \vee \neg Q$: is equivalent to saying I will not eat strawberry ice or I will not eat chocolate ice cream
2. $\neg (P \vee Q)$ is equivalent to $\neg P \wedge \neg Q$
$\neg (P \vee Q)$: is equivalent to saying: I will not eat either strawberry ice cream or chocolate ice cream
$\neg P \wedge \neg Q$: is equivalant to saying I will not eat strawberry ice cream and I will not eat chocolate ice cream

Having drawn these analogies I'm at a loss to see how the two laws differ.
Is it possible using the same analogies someone can clarify the finer distinctions between the two laws. Thank you.
 A: 

*"I will not eat both strawberry and chocolate ice cream" in the intended sense,
i.e. "It is not the case that I will eat both strawberry and chocolate ice cream" 


is obviously distinct from 


*"I will not eat strawberry ice cream and I will not eat chocolate ice cream". 


Suppose I eat strawberry but not chocolate. 3 then is true and 4 false.
3 and 4  show that (1a) $\neg(P \land Q)$ is distinct from (2a) $\neg P \land \neg Q$



*"I will not eat strawberry ice or I will not eat chocolate ice cream" 


is also obviously distinct from 


*"I will not eat either strawberry ice cream or chocolate ice cream". 


Suppose I don't eat strawberry but do eat chocolate. Then 3 is true and 4 false.
5 and 6 show that (1b) $\neg P \lor \neg Q$ is distinct from (2b) $\neg (P \lor Q)$.

Hence the  claim (1), that 1a is equivalent to 1b, is indeed a distinct claim from the claim (2), that 2a is equivalent to 2b.
A: The statements are equivalent.  Start with
$$\lnot (P \land Q) = \lnot P \lor \lnot Q$$
Substitute $\lnot A = P$ and $\lnot B = Q$
$$\lnot (\lnot A \land \lnot B) = \lnot \lnot A \lor \lnot \lnot B = A \lor B$$
Negate both sides:
$$\lnot A \land \lnot B = \lnot (A \lor B)$$
Which is the second Demorgan's law.
