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# MCQs on Predicate Logic Quantifiers

. Let P (x) denote the statement “x >7.” Which of these have truth value true?

a) P (0)

b) P (4)

c) P (6)

d) P (9)

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Answer: d

Explanation: Put x=9, 9>7 which is true.

Let Q(x) be the statement “x < 5.” What is the truth value of the quantification ∀xQ(x), having domains as real numbers.

a) True

b) False

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Answer: b

Explanation: Q(x) is not true for every real number x, because, for instance, Q(6) is false. That is, x = 6 is a counterexample for the statement ∀xQ(x). This is false.

. Determine the truth value of ∀n(n + 1 > n) if the domain consists of all real numbers.

a) True

b) False

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Answer: a

Explanation: There are no elements in the domain for which the statement is false.

Let P(x) denote the statement “x = x + 7.” What is the truth value of the quantification ∃xP(x), where the domain consists of all real numbers?

a) True

b) False

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Answer: b

Explanation: Because P(x) is false for every real number x, the existential quantification of Q(x), which is ∃xP(x), is false.

Let R (x) denote the statement “x > 2.” What is the truth value of the quantification ∃xR(x), having domain as real numbers?

a) True

b) False

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Answer: a

Explanation: Because “x > 2” is sometimes true—for instance, when x = 3–the existential quantification of R(x), which is ∃xR(x), is true.

The statement,” Every comedian is funny” where C(x) is “x is a comedian” and F (x) is “x is funny” and the domain consists of all people.

a) ∃x(C(x) ∧ F (x))

b) ∀x(C(x) ∧ F (x))

c) ∃x(C(x) → F (x))

d) ∀x(C(x) → F (x))

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Answer: d

Explanation: For every person x, if comedian then x is funny.

The statement, “At least one of your friends is perfect”. Let P (x) be “x is perfect” and let F (x) be “x is your friend” and let the domain be all people.

a) ∀x (F (x) → P (x))

b) ∀x (F (x) ∧ P (x))

c) ∃x (F (x) ∧ P (x))

d) ∃x (F (x) → P (x))

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Answer: c

Explanation: For some x, x is friend and funny.

”Everyone wants to learn cosmology.” This argument may be true for which domains?

a) All students in your cosmology class

b) All the cosmology learning students in the world

c) Both of the mentioned

d) None of the mentioned

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Answer: c

Explanation: Domain may be limited to your class or may be whole world both are good as it satisfies universal quantifier.

Let domain of m includes all students, P (m) be the statement “m spends more than 2 hours in playing polo”. Express ∀m ¬P (m) quantification in English.

a) A student is there who spends more than 2 hours in playing polo

b) There is a student who does not spend more than 2 hours in playing polo

c) All students spends more than 2 hours in playing polo

d) No student spends more than 2 hours in playing polo

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Answer: d

Explanation: There is no student who spends more than 2 hours in playing polo.

Determine the truth value of statement ∃n (4n = 3n) if the domain consists of all integers.

a) True

b) False

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Answer: a

Explanation: For n=0, 4n=3n hence, it is true.