The puzzle I am working on is:
"Let $p$, $q$, and $r$ be the propositions
$p$: Grizzly bears have been seen in the area.
$q$: Hiking is safe on the trail.
$r$: Berries are ripe along the trail.
Write these propositions using $p$, $q$, and $r$ and logical
connectives (including negations).
a) Berries are ripe along the trail, but grizzly bears have
not been seen in the area.
b)Grizzly bears have not been seen in the area and hiking on the trail is safe, but berries are ripe along the
trail.
c) If berries are ripe along the trail, hiking is safe if and
only if grizzly bears have not been seen in the area.
d)It is not safe to hike on the trail, but grizzly bears have
not been seen in the area and the berries along the trail
are ripe.
e) 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.
f) Hiking is not safe on the trail whenever grizzly bears
have been seen in the area and berries are ripe along
the trail.
The only one that I bewilder by is e). The answer I came by was $(\neg p \wedge \neg r) \implies q$. However, the true answer is, "$(q→(¬r∧¬p))∧¬((¬r∧¬p)→q)$". Would someone be so gracious as to explain to me why this is the answer?
Best Answer
e) 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.
$(1)$ For hiking to be safe ($a:=q$), the necessary condition ($b:= \lnot r \land \lnot p$) in your case is that berries not be ripe and for grizzly bears to not to have been seen in the area. So we have
$(2)$ However, you are explicitly told that this condition: ($\lnot r \land \lnot p$) is not sufficient, so you have to negate the converse of $(1)$: you need to negate $(\lnot r \land \lnot p) \rightarrow q$. This gives us:
To write the complete statement, you need the connective $\land$ in between $(1)$ and $(2)$:
$$[q\rightarrow(\lnot r \land \lnot p)]\land \lnot[(\lnot r \land \lnot p) \rightarrow q].$$
I agree with all that the translation used by the text is terrible, as it oversimplifies matters. But what the text was trying to convey is described above, in $(2)$. (No wonder you're confused by the negated implication.)