Negating "If no one is absent, then if the weather permits, we will study outside"

Question

I am a beginner; please help solve this.

Write the negation of the statement:

"If no one is absent, then if the weather permits, we will study outside."

Answer 1

First, write the statement as a logical statement (with appropriate notation).

A: "No one is absent."

W: "Weather permits"

S: "We will study outside."

Then we have the proposition, symbolically: $$A\rightarrow(W \rightarrow S)$$

Then, since we want to negate the proposition, we work with:

$$\begin{align} \lnot\Big(A\rightarrow(W \rightarrow S)\Big)& \equiv \lnot\Big(\lnot A \lor (\lnot W \lor S)\Big)\tag{$p\rightarrow q \equiv \lnot p \lor q$}\\ \\ & \equiv A \land W \land \lnot S\tag{DeMorgan's}\end{align}$$

So the negation of the statement translated back to English gives us:

"No one is absent and weather is permitting, but (and) we won't study outside."

Answer 2

p="no one is absent"
q="the weather permits"
r="we will study outside"
$$\neg(p\to(q\to r))$$ $$\neg(\neg p \vee (q\to r))$$ $$(\neg \neg p) \wedge \neg(q\to r)$$ $$p \wedge \neg(\neg q \vee r)$$ $$p\wedge ((\neg \neg q)\wedge (\neg r))$$ $$p\wedge q\wedge (\neg r)$$

No one is absent and the weather permits and we will not study outside.

Answer 3

The negation of $A\to B$ is $A\land\neg B$, so

No one is absent and not (if the wheather permits, we will study outside)

Do the same for the second part to obtain

No one is absent and the wheather permits and we will not study outside.

If you like, replace "no one is absent" with "everybody is present".