Module 6 Homework Assignment

Assuming that the domain of discourse is the set of real numbers, select the expression that represents the given mathematical sentence. The product of two negative numbers is a positive number. ∀x ∀y ((x > 0 ˄ y > 0) → (xy > 0)) ∀x ∀y ((x > 0 ˅ y > 0) → (xy > 0)) ∀x ∀y ((x > 0 ˄ y > 0) ˄ (xy > 0)) ∀x ∀y ((x < 0 ˄ y < 0) → (xy > 0))

∀x ∀y ((x < 0 ˄ y < 0) → (xy > 0))

Consider the following statement. How would you express it using quantifiers? For all real numbers x and for all real numbers y there is a real number z such that x + y = z. ∀x ∀y ∃z (x + y = z) ∀xy ∃z (x + y = z) ∃z ∀x ∀y (x + y = z) ∀x ∃y ∃z (x + y = z)

∀x ∀y ∃z (x + y = z)

Consider the FOL formula ∀x (P(x, a) ˄ Q(x, b)) where P, Q are relations and a, b are constants. Which options represent well-defined interpretations of the formula? Select all that apply. 1. a = 3, b = 2, P = {(5, 3), (4, 3), (2, 2)}, Q = {(1, 2), (3, 2), (5, 2)} 2. Domain of discourse = {Red, Green, Orange, Blue}, a = Red, b = Orange, P = {(Orange, Red), (Blue, Blue), (Blue, Red)}, Q = {(Red, Orange), (Green, Orange), (Green, Blue)} 3. Domain of discourse = {2, 3, 4, 5}, a = 3, b = 4, x = 3, P = {(2, 3), (3, 4), (4, 3)}, Q = {(2, 4), (4, 3), (5, 4)} 4. Domain of discourse = {1, 2, 3}, a = 1, b = 4, P = {(1, 1), (2, 1)}, Q = {(3, 4)}

2

In the options of this question, several FOL formulas are given. Consider the interpretation { domain of discourse = set of all real numbers, P = {(x, y, z) | (x + y) / 2 = z} (i.e. P is the set of 3-tuples such that the third component is the average of the first and second), a = 75b = 25 } Which formulas would be true (T) under the given interpretation? Select all that apply. 1. ∀n P(a, b, n) 2. ∃n P(a, b, n) 3. ∀n P(a, n, b) 4. ∃n P(a, n, b)

2 and 4

In which expressions the quantifiers (the universal, the existential, or both) are NOT correctly used? Select all that apply. Note that the question refers to the syntax of the expressions, not their truth values; assume that the domain is the set of real numbers. 1. ∀x ∀y ((x = -1/y ˄ x ≠ 0) → (x > y)) 2. ∀x ∃yz ((x = y ˄ y = z) → (x = z)) 3. ∃x ∀y ∃z (x + y + z = -x - y - z) 4. ∀xy (x + y = ∃z)

2 and 4

Which of the following relations define functions? Select all that apply. 1. Consider the set S = {1, 2, 3} and let R be the relation on S × S, R = {(1, 1), (2, 3), (1, 3)}. 2. Consider the sets S1= {1, 2, 3, 4} and S2 = {3, 4, 5} and let R be the relation on S1 × S2, R = {(1, 3), (2, 5), (4, 5)}. 3. Let R1 be the relation on ℤ × ℤ, R1 = {(x,y) | x ϵ ℤ and x < y}, where ℤ denotes the set of integers. 4. Consider the set S = {1, 2, 3, 4} and let R be the relation on S × S, R = {(1, 1), (2, 3), (4, 3)}.

2 and 4

Let the domain of discourse of variables x and y be the set of real numbers. Which propositions have a truth value of T? Select all that apply. 1.∀x ∃y (x = y2) 2.∀x ∃y (x2 = y) 3.∀x ∃y (y < x) 4.∀x ∃y (x ≤ y)

2, 3, and 4

Let the domain of the variable x be the set of real numbers. What is the truth value of the proposition? ∀x (x2 < x)

False

Let the domain of the variable x be the set of real numbers. What is the truth value of the proposition? ∀x (x2 ≤ x)

False

Let the domain of the variable x be the set of real numbers. What is the truth value of the proposition? ∃x (x2 < x)

True

Let the domain of the variable x be the set of real numbers. What is the truth value of the proposition? ∃x (x2 ≤ x)

True

Let the domain of discourse be the set of all people. Consider the predicates: employed(x): x is employed worksFromHome(x): x works from home hasFamily(x): x has a family hardWorking(x): x is hardworking Select the expression that represents the sentence: There is a hardworking employee who does not work from home and has a family. ∃x (employed(x) → (hardWorking(x) ˄ ¬worksFromHome(x) ˄ hasFamily(x))) ∃x ((hardWorking(x) ˅ employed(x)) ˄ ¬worksFromHome(x) ˄ hasFamily(x)) ∃x (hardWorking(x) ˄ employed(x) ˄ ¬worksFromHome(x) ˄ hasFamily(x)) ∃x ((hardWorking(x) ˄ employed(x)) → (¬worksFromHome(x) ˄ hasFamily(x)))

∃x (hardWorking(x) ˄ employed(x) ˄ ¬worksFromHome(x) ˄ hasFamily(x))

In the given options, P(x, y) denotes the statement "x knows y" and the domain of discourse is the set of all persons. Select the expression that represents the sentence: There is at least one person that everybody knows. ∃y ∀x P(x, y) ∀x ∃y P(x, y) ∃x ∃y P(x, y) ∀x ∀y P(x, y)

∃y ∀x P(x, y)