Answer to Question #140471 in Discrete Mathematics for umirzoq

Question

1) 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.

Answer 1

$P$ : Grizzly bears have been seen in the area.

$Q$ : Hiking is safe on the trail.

$R$ : Berries are ripe along the trail.

Various English words and word pairs express logical connectives, and some of them are synonymous. These include, among others:

The word "not" correspond to the operation negation "$\neg$".

The words "and", "but" correspond to the operation conjunction "$\land$".

The word pair "if ... then" corresponds to the operation material implication "$\Rightarrow$".

The phrase "if and only if" corresponds to the operation biconditional "$\Leftrightarrow$".

Therefore, we have the following logical formulas.

a) $R\land \neg P$ corresponds to the proposition "Berries are ripe along the trail, but grizzly bears have not been seen in the area".

b) $\neg P\land Q\land R$ corresponds to the proposition "Grizzly bears have not been seen in the area and hiking on the trail is safe, but berries are ripe along the trail".

c) $R \Rightarrow(Q\Leftrightarrow \neg P)$ corresponds to the proposition "If berries are ripe along the trail, hiking is safe if and only if grizzly bears have not been seen in the area".