Logics – Nested Quantifiers

1. Let Q(x, y) denote “M + A = 0.” What is the truth value of the quantifications AM Q(M, A).
a) True
b) False

Explanation: For each A there exist only one M, because there is no real number A such that M + A = 0 for all real numbers M.

2. Translate xy(x < y) in English, considering domain as a real number for both the variable.
a) For all real number x there exists a real number y such that x is less than y
b) For every real number y there exists a real number x such that x is less than y
c) For some real number x there exists a real number y such that x is less than y
d) For each and every real number x and y such that x is less than y

3. “The product of two negative real numbers is not negative.” Is given by?
a) ∃x ∀y ((x < 0) ∧ (y < 0) → (xy > 0))
b) ∃x ∃y ((x < 0) ∧ (y < 0) ∧ (xy > 0))
c) ∀x ∃y ((x < 0) ∧ (y < 0) ∧ (xy > 0))
d) x y ((x < 0) (y < 0) → (xy > 0))

4. Let Q(x, y) be the statement “x + y = x − y.” If the domain for both variables consists of all integers, what is the truth value of xQ(x, 4).
a) True
b) False

5. Let L(x, y) be the statement “x loves y,” where the domain for both x and y consists of all people in the world. Use quantifiers to express, “Joy is loved by everyone.”
a)
x L(x, Joy)
b) ∀y L(Joy,y)
c) ∃y∀x L(x, y)
d) ∃x ¬L(Joy, x)

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6. Let T (x, y) mean that student x likes dish y, where the domain for x consists of all students at your school and the domain for y consists of all dishes. Express ¬T (Amit, South Indian) by a simple English sentence.
a) All students does not like South Indian dishes.
b) Amit does not like South Indian people.
c) Amit does not like South Indian dishes.
d) Amit does not like some dishes.

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7. Express, “The difference of a real number and itself is zero” using required operators.
a) ∀x(x − x! = 0)
b) x(x − x = 0)
c) ∀x∀y(x − y = 0)
d) ∃x(x − x = 0)

8. Use quantifiers and predicates with more than one variable to express, “There is a pupil in this lecture who has taken at least one course in Discrete Maths.”
a)
xyP (x, y), where P (x, y) is “x has taken y,” the domain for x consists of all pupil in this class, and the domain for y consists of all Discrete Maths lectures
b) ∃x∃yP (x, y), where P (x, y) is “x has taken y,” the domain for x consists of all Discrete Maths lectures, and the domain for y consists of all pupil in this class
c) ∀x∀yP(x, y), where P (x, y) is “x has taken y,” the domain for x consists of all pupil in this class, and the domain for y consists of all Discrete Maths lectures
d) ∃x∀yP(x, y), where P (x, y) is “x has taken y,” the domain for x consists of all pupil in this class, and the domain for y consists of all Discrete Maths lectures

9. Determine the truth value of nm(n + m = 5 n − m = 2) if the domain for all variables consists of all integers.
a) True
b) False

10. Find a counterexample of ∀x∀y(xy > y), where the domain for all variables consists of all integers.
a) x = -1, y = 17
b) x = -2 y = 8
c) Both x = -1, y = 17 and x = -2 y = 8
d) Does not have any counter example