Question

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.

Short answer

a)$r\wedge \neg p$

b)$\neg p\wedge q\wedge r$

c)$(r\to q)\leftrightarrow \neg p$

d)$\neg q\wedge \neg p\wedge r$

e)$q\to (\neg r\wedge p)$

f)$(p\wedge r)\to \neg q$

Step-by-step solution

Definition of proposition

Proposition:

1. It is a declarative statement which can be either true or false.

2. It cannot be both true and false simultaneously

Proposition using p,q and r for a)

p: Grizzly bears have been seen in the area.

q: Hiking is safe on the trail.

r: Berries are ripe along the trail.

Berries are ripe along the trail, but grizzly bears have not been seen in the area.

The above given statement means $r\wedge \neg p$.

Step 3:Proposition using p,q and r for b)

Grizzly bears have not been seen in the area and hiking on the trail is safe, but berries are ripe along the trail.

The given statement means $\neg p\wedge q\wedge r$.

Proposition using p,q and r for c)

If berries are ripe along the trail, hiking is safe if and only if grizzly bears have not been seen in the area.

The above given statement means $(r\to q)\leftrightarrow \neg p$.

Proposition using p,q and r  for 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.

The given statement means $\neg q\wedge \neg p\wedge r$.

Proposition using p,q and r for 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

The given statement means $q\to (\neg r\wedge p)$.

Proposition using p,q and r for 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 given statement means $(p\wedge r)\to \neg q$.