4. (a) [4 points] Let p and q and r be the following propositions:
p: Mary does every homework.
q: Mary passes this course.
r: Mary misses the final exam.
Write the following propositions using p, q, and r and negations and logical connectives:
(i) Mary passes this course, but she does not do every homework.
(ii) If Mary either misses the final exam or does not do every homework, then she does not pass this course.
(iii) It is not the case that Mary both passes this course and also does every homework. (
iv) Mary passes this course if and only if she does every homework.
(b) [3 points] Let p. q and r be the following propositions:
p: Bob has been elected.
q: The votes have been counted.
r: The election is decided.
Express each of the following propositions as an English sentence:
(i) ~r ^ p
(ii) q → p
(iii) r→ (p^q)
(e) [3 points] State the following logical equivalences using propositional variables p, q, r and negations and logical connectives.
(i) Conjunction is associative.
(ii) Disjunction is distributive over conjunction.
(iii) De Morgan's law for the negation of a disjunction of three propositions p, q, r.