Question

Let p and q be the propositions “Swimming at the New Jersey shore is allowed” and “Sharks have been spotted near the shore,” respectively. Express each of these compound propositions as an English sentence

a) $\neg q$

b) $p\wedge q$

c)$\neg p\vee q$

d)$p\to \neg q$

e)$\neg q\to p$

f )$\neg p\to \neg q$

g)$p\leftrightarrow \neg q$

h)$\neg p\wedge (p\vee \neg q)$

Short answer

1. Sharks have not been spotted near the shore.
2. Swimming at the New Jersey shore is allowed and sharks have been spotted near the shore.
3. Swimming at the New Jersey shore is not allowed or sharks have been spotted near the shore.
4. If swimming at the New Jersey shore is allowed then sharks have not been spotted near the shore.
5. If sharks have not been spotted near the shore then swimming at the New Jersey shore is allowed.
6. If Swimming at the New Jersey shore is not allowed then sharks have not been spotted near the shore.
7. Swimming at the New Jersey shore is allowed if and only if sharks have not been spotted near the shore.
8. Swimming at the New Jersey shore is not allowed and swimming at the New Jersey shore is allowed or sharks have not been spotted near the shore

Step-by-step solution

Definition of proposition

Proposition:
1. It is a declarative statement that can be either true or false.
2. It cannot be both true and false simultaneously

English sentence for a)

q: “Sharks have been spotted near the shore”,

$\neg q$would mean,

Sharks have not been spotted near the shore.

English sentence for b)

We write given statement as,

$p\wedge q$would mean,

Swimming at the New Jersey shore is allowed and sharks have been spotted near the shore.

English sentence for c)

p: Swimming at the New Jersey shore is allowed.

q: Sharks have been spotted near the shore,

$\neg p\vee q$would mean,

Swimming at the New Jersey shore is not allowed or sharks have been spotted near the shore

English sentence in d)

We write the given statement as,

$p\to \neg q$would mean,

If swimming at the New Jersey shore is allowed then sharks have not been spotted near the shore

English sentence in e)

We write given statement as,

$\neg q\to p$would mean,

If sharks have not been spotted near the shorethen swimming at the New Jersey shore is allowed

English sentence in f)

We write given statement as,

$\neg p\to \neg q$would mean,

If swimming at the New Jersey shore is not allowed then sharks have not been spotted near the shore

English sentence in g)

We write given statement as,

$p\leftrightarrow \neg q$would mean,

Swimming at the New Jersey shore is allowed if and only if sharks have not been spotted near the shore.

English sentence in h)

We write given statement as,

$\neg p\wedge (p\vee \neg q)$would mean,

Swimming at the New Jersey shore is not allowed and swimming at the New Jersey shore is allowed or sharks have not been spotted near the shore.