**1. Let Q(x, y) denote “M + A = 0.” What is the truth value of the quantifications ****∃****A****∀****M 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 ****∀****x****∃****y(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)

.

**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.**

.

**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) **

**∃**

**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**

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 ****∃****n****∃****m(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