In the following question, the domain of discourse is a set of male patients in a clinical study. Define the following predicates:

P(x): x was given the placebo D(x): x was given the medication A(x): x had fainting spells M(x): x had migraines

Suppose that there are five patients who participated in the study. The table below shows the names of the patients and the truth value for each patient and each predicate: P(x) D(x) A(x) M(x) Frodo T F F T Gandalf F T F F Gimli F T T F Aragorn T F T T Bilbo T T F F

For each of the following quantified statements, indicate whether the statement is a proposition. If the statement is a proposition, give its truth value and translate the expression into English.

(a)

x (M(x) D(x))

(b)

x M(x) x D(x)

(c)

x M(x) D(x)

(d)

x (A(x) M(x))