Solution to Question 17 Consider the following proposition: For any predicates P(x) and Q(x) over a domain … - Sikademy
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Archangel Macsika

Question 17 Consider the following proposition: For any predicates P(x) and Q(x) over a domain D, the negation of the statement ∃x ∈ D, P(x) ∧ Q(x) is the statement ∀x ∈ D, P(x) → ¬Q(x). We can use this truth to write the negation of the following statement: “There exist integers a and d such that a and d are negative and a/d = 1 + d/a.” Which one of the alternatives provides the negation of this statement? 1. There exist integers a and d such that a and d are positive and a/d = 1 + d/a. 2. For all integers a and d, if a and d are positive then a/d  1 + d/a. 3. For all integers a and d, if a and d are negative then a/d  1 + d/a. 4. For all integers a and d, a and d are positive and a/d  1 + d/a.

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Let P(x) be the statement “ a and d are negative" and Q(x) be the statement “\frac{a}{d} = 1 + \frac{d}{a} ”. Then the statement “There exist integers a and d such that a and d are negative and \frac{a}{d} = 1 + \frac{d}{a} .” is ∃x ∈ D, P(x) ∧ Q(x) and its negation is the statement ∀x ∈ D, P(x) → ¬Q(x), that is

the statement "For all integers a and d, if a and d are negative then \frac{a}{d} \ne 1 + \frac{d}{a} ."


Answer: 3


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Question ID: mtid-5-stid-8-sqid-2840-qpid-1397