# Negation of a Logical Statement; Proper English Translation

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?

## 1 Answer

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, $$\lnot (p\lor q)\equiv\lnot p\land\lnot q.$$

* : 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.