Let p and q be the propositions "The election is decided" and "The votes have been counted," respectively. Express each of these compound
propositions as English sentences.
a) ¬ p : The election is not (yet) decided.
b) p ∨ q : The election is decided or the votes have been counted.
c) ¬ p∧q : The votes have been counted but the election is not (yet)
decided.
d) q → p : If the votes are counted then the election is decided.
e) ¬ q → ¬ p : The election is not decided unless the votes have
been counted.
f) ¬ p → ¬ q : The votes have not been counted unless the election
has been decided. This is equivalent to proposition d).
g) p ↔ q : The election is decided if and only if the votes have been
counted.
h) ¬ q ∨ (¬ p ∧ q) : The votes have not been counted, or they have
been counted but the election is not (yet) decided.