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) ¬q b) p ∧ q c) ¬p ∨ q d) p → ¬q e) ¬q → p f) ¬p → ¬q g) p ↔ ¬q h) ¬p ∧ (p ∨ ¬q)

Answer 1

Answer with explanation:

Given propositions: p ="Swimming at the New Jersey shore is allowed"

q = "Sharks have been spotted near the shore"

We need to express each of these compound propositions as an English sentence.

a) ¬q : not q

Sentence: Sharks have not been spotted near the shore.

b) p ∧ q : p and q

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

c) ¬p ∨ q  : not p or q.

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

d) p → ¬q  : If p then not q.

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

e) ¬q → p : If not q then p.

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

f) ¬p → ¬q : If not p then not q.

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

g) p ↔ ¬q : p if and only if not q

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

h) ¬p ∧ (p ∨ ¬q) : not p and, p or not q.

Sentence: 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.