Question

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.)

Answer 1

Answer:

Explanation:

To begin, i will like to break this down to its simplest form to make this as simple as possible.

Let us begin !

Here statement can be translated as: ∃x (P(x) ∧ M(x))

we require the Negation: ¬∃x (P(x) ∧ M(x))

De morgan's law can be stated as:

1) ¬(a ∧ b) = (¬a ∨ ¬b)

2) ¬(a v b) = (¬a ∧ ¬b)

Also, quantifiers are swapped.

Applying De Morgan's law, we have: ∀x (¬P(x) ∨ ¬M(x)) (∃ i swapped with ∀ and intersecion is replaced with union.)

This is the translation of above

English: Every patient was either not given the placebo or did not have migrane(or both).

cheers i hope this helped !!