[Math] Necessary but not sufficient in logic

discrete mathematicslogicpropositional-calculus

I am working through sample questions and am having a bit of trouble understanding the solution.

Write using logical connectives:

p : Grizzly bears have been seen in the area.

q : Hiking is safe on the trail.

r : Berries are ripe along the trail.

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 came up with

$$q \rightarrow(\lnot r \land \lnot p)$$

However, the solution is

$$q \rightarrow (\lnot r \land \lnot p) \land \lnot((\lnot r\land \lnot p)\rightarrow q)$$

I am wondering why the attached part is required. I guess my problem would be with understanding the question at hand. Would the last two propositions be sufficient enough for the first proposition to be true, are they both together still insufficient?

Any help is appreciated! Thank you.

Best Answer

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.

$$(\underbrace{(\lnot r \land \lnot p)\leftarrow q}_{\text{it is neccessary}}) \underbrace{\land}_{\text{and/but}} (\underbrace{\lnot((\lnot r\land \lnot p)\rightarrow q)}_{\text{it is not sufficient}})$$

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