Negation of a Logical Statement; Proper English Translation

Question

Consider the following two propositions:

• $p$: We can go to Cancun.
• $q$: We can go to Iceland.

Using symbolic notation,

a) Form the conjunction ($\land$).

$p \land q$: We can go to Cancun and we can go to Iceland.

b) Form the disjunction ($\lor$).

$p \lor q$: We can go to Cancun or we can go to Iceland.

c) Write the negation ($\neg$) of part a) as a logical statement and as an English sentence.

$\neg p \land \neg q$: We cannot go to Cancun and we cannot go to Iceland.

d) Write the negation ($\neg$) of part b) as a logical statement and as an English sentence.

$\neg p \lor \neg q$: We cannot go to Cancun or we cannot go to Iceland.

I just want to make sure that my answers are correct. Specifically, I am worried about (d), as I find it to be confusing; is it correct?

Answer 1

You should recheck $\tt c)$ and $\tt d)$. The negation of $p\land q$ isn't $\lnot p\land\lnot q$, rather one has:* $$\lnot (p\land q)\equiv\lnot p\lor\lnot q.$$ Similarly, $$\lnot (p\lor q)\equiv\lnot p\land\lnot q.$$

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^{* : The reason behind that is that the proposition $p\land q$ is false not only when both $p$ and $q$ are false (i.e. when $\lnot p\land\lnot q$ is true), but also when either one of them is false (i.e. when $\lnot p\lor\lnot q$ is true). A similar line of reasoning can be given for the second equivalence.}