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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. 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.

          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.
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.

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Added by Angela C.

Discrete Mathematics and its Applications

Kenneth Rosen 8th Edition

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Solved by Expert Manisha Sarker

11/09/2021

Step 1

” q: “Hiking is safe on the trail.” r: “Berries are ripe along the trail.” Note that “not” of a proposition is written as ¬ (e.g., ¬p means “grizzly bears have not been seen in the area”) and “but” functions like “and.” Show more…

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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. 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.

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Transcript

00:01 In this problem, we have been given three propositions, p, q and r.

00:05 Key is the proposition grizzly bears have been seen in the area.

00:08 Q is the proposition hiking is safe on the trail and r is the proposition berries are ripe along the trail.

00:15 We need to use these to translate the given statements.

00:20 So the first statement is berries are ripe along the trail but grizzly bears have not been seen in the area.

00:25 So that means that berries are ripe along the trail and grizzly bears have not been seen in the area.

00:31 So berries are ripe along the trail that is r.

00:34 This is the symbol for and.

00:37 And we have grizzly bears have not been seen in the area.

00:40 Grizzly bears have been seen in the area is p.

00:43 So grizzly bears have not been seen in the area is not p.

00:46 So the answer is r and not p.

00:50 The next statement is grizzly bears have not been seen in the area and hiking on the trail is safe, but berries are ripe along the trail.

00:58 So this means that grizzly bears have not been seen in the area, which is not p, and hiking on the trail is safe, so that's q, and berries are ripe along the trail.

01:11 So that is r.

01:12 So not p and q and r.

01:15 The next statement is if berries are ripe along the trail, hiking is safe if and only if grizzly bears have not been safe seen in the area.

01:23 So if berries are ripe along the trail.

01:26 So that means if r then, so that means r implies.

01:31 Hiking is safe if and only if grizzly bears have not been seen in the area.

01:36 So that means q, if and only if not p.

01:42 So we have r implies q and the biconditional, then not p.

01:50 Then we have the next statement.

01:52 It is not safe to hike on the trail, but grizzly bears have not been seen in the area.

01:56 And the bearers the berries along the trail are ripe.

01:59 So it is not safe to hike along the trail...

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