This is from Discrete Mathematics and Its Applications:
Let $p, q,$ and $r$ be the propositions:
$\quad p:$ Grizzly bears have been seen in the area.
$\quad q:$ Hiking is safe on the trail.
$\quad r:$ Berries are ripe along the trail.
Write these propositions using $p,q,$ and $r$ and logical connectives (including negation):
- For hiking on the trail to be safe, it is necessary but not sufficient that berries not be ripe along the trail and for grizzly bears not to have been seen in the area.
I read up on necessary and sufficient from here What is the difference between necessary and sufficient conditions?
- If $p \to q$ ($p$ implies $q$), then $p$ is a sufficient condition for $q$.
- If $\neg p \to \neg q$ (not $p$ implies not $q$), then $p$ is a necessary condition for $q$.
From these two conditions how would you apply necessary but not sufficient?
The way I expressed this is: $$ (\neg r \land \neg p) \to q$$
I mainly got to this because "if $p$ then $q$" is the same as $q$ is necessary for $p$. $q$ in this case would be two conditions — berries not be ripe along the trails and grizzly bears not to have been seen in the area".
How would the necessary but not sufficient clause affect the answer though? Would it make a difference?