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.
The English sentence of each compound proposition is discussed below.
Given propositions,
p is "Swimming at the New Jersey shore is allowed".
And q is "Sharks have been spotted near the shore".
We have to express each of these compound propositions as an English sentence.
a. ¬q means not q.
Since,q is "Sharks have been spotted near the shore".
So Sentence becomes, Sharks have not been spotted near the shore.
b. p ∧ q means we have to use and to combine both the propositions.
So sentence becomes, Swimming at the New Jersey shore is allowed and sharks have been spotted near the shore.
c. ¬p ∨ q means not p or q.
So sentence becomes, Swimming at the New Jersey shore is not allowed or sharks have been spotted near the shore.
d. p → ¬q means If p then not q.
So sentence becomes, If swimming at the New Jersey shore is allowed then sharks have not been spotted near the shore.
e. ¬q → p means If not q then p.
So sentence becomes, If sharks have not been spotted near the shore then swimming at the New Jersey shore is allowed .
f. ¬p → ¬q means If not p then not q.
So sentence becomes, If swimming at the New Jersey shore is not allowed then sharks have not been spotted near the shore.
g. p ↔ ¬q means p if and only if not q.
So sentence becomes, Swimming at the New Jersey shore is allowed if and only if sharks have not been spotted near the shore.
h. ¬p ∧ (p ∨ ¬q) means not p and, p or not q.
So sentence becomes, 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.
Hence the sentences as a compound proposition is discussed above.
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