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MODULE ONE PROBLEM SET
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Jaime Rowland
snhu
Directions: Type your solutions into this document and be sure to show all steps for arriving at your solution. Just giving a final number may not receive full credit.
PROBLEM 1
In the following question, the domain of discourse is a set of male patients in a clinical study. Define the following predicates:
P() : was given the placebo
: D(r) : x was given the medication
: M(r) : x had migraines
Translate each of the following statements 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.
:3x(P(x) A D(x))
: Negation: -3x (P(x) A D(x))
: Applying De Morgan's law: Vx (-P(x) V -D(x))
: English: Every patient was either not given the placebo or not given the medication (or both).
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(a) Every patient was given the medication or the placebo or both
Logical Expression:- (x) (P(x) V D(x)) Negation:- (x) (P(x) V D(x)) (x) (- P(x) - D(x)) English Meaning There exists some patient who was not given both placebo and medica- tion.
(b) Every patient who took the placebo had migraines. (Hint: you will need to apply the conditional identity, p q = -p V q.)
Logical Expression:- (x)(P(x)M(x)) Negation:- (x) (P(x) M(x)) - (x) (- P(x) M(x)) (x) (P(x)-M(x)) English Meaning: There exists some patient who was given placebo and was not having migraines.
(c) There is a patient who had migraines and was given the placebo
Logical Expression:- (x) (M(x) P(x)) Negation:- - (x) (M(x) P(x)) (x) (-M(x) -P(x)) (x) (M(x) -P(x)) English Meaning: Every patient who had migraines was not given pl