Clarification needed regarding DeMorgan's Law

Question

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

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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

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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.

Answer 1

3. "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

4. "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) $\neg P \land \neg Q$

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5. "I will not eat strawberry ice or I will not eat chocolate ice cream"

is also obviously distinct from

6. "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) $\neg (P \lor Q)$.

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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.

Answer 2

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.