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...

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
• M(x): x had migraines

Translate each statement into a logical expression. Then negate the expression by adding a negation operation to the beginning of the expression. Apply De Morgan's law until each negation operation applies directly to a predicate and then translate the logical expression back into English.

Sample question: Some patient was given the placebo and the medication.

• ∃x (P(x) ∧ D(x))
• Negation: ¬∃x (P(x) ∧ D(x))
• Applying De Morgan's law: ∀x (¬P(x) ∨ ¬D(x))
• English: Every patient was either not given the placebo or not given the medication (or both).

(a)

Every patient was given the medication.

(b)

Every patient was given the medication or the placebo or both.

(c)

There is a patient who took the medication and had migraines.

(d)

Every patient who took the placebo had migraines. (Hint: you will need to apply the conditional identity, p → q ≡ ¬p ∨ q.)

(e)

There is a patient who had migraines and was given the placebo.