Question

1.   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

Let us express each of these compound propositions as an English sentence.

a) The proposition ¬q  as an English sentence is "Sharks have not been spotted near the shore"

b) The proposition p ∧ q   as an English sentence is "Swimming at the New Jersey shore is allowed and sharks have been spotted near the shore"

c) The proposition ¬p ∨ q   as an English sentence is "Swimming at the New Jersey shore is not allowed or sharks have been spotted near the shore"

d) The proposition p →¬q   as an English sentence is "If swimming at the New Jersey shore is allowed then sharks have not been spotted near the shore"  

e) The proposition ¬q → p as an English sentence is "If sharks have not been spotted near the shore then swimming at the New Jersey shore is allowed"

f ) The proposition ¬p →¬q as an English sentence is "If swimming at the New Jersey shore is not allowed then sharks have not been spotted near the shore"

g) The proposition p ↔¬q  as an English sentence is "Swimming at the New Jersey shore is allowed if and only if sharks have not been spotted near the shore" 

h) The proposition ¬p ∧ (p∨ ¬q) as an English sentence is "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"