4. State the value of x after the statement if P(x) then x := 1 is executed, where P(x) is the statement “x > 1,” if the value of x when this statement is reached is a. x=0
i. x is equal to zero, the condition is false.
ii. x is equal to one, the condition is false.
iii. x is equal to two, the condition is true. So the statement x: = 1 is executed. 6. Let N(x) be the statement “x has visited North Dakota,” where the domain consists of the students in your school. Express each of these quantifications in English. a. ∃xN(x)
i. There exists a student in school, who has visited North Dakota. b. ∀xN(x)
ii. All students in the school have visited North Dakota c. ¬∃xN(x)
iii. There does NOT exist a student in the school who has visited North Dakota d. ∃x¬N(x)
iv. There exists a student in school, who has NOT visited North Dakota. e. ¬∀xN(x)
v. Not all students in the school have visited North Dakota. f. ∀x¬N(x)
vi. All students in the school have NOT visited North Dakota. (no student has visited North Dakota) 10. Let C(x) be the statement “x has a cat,” let D(x) be the statement “x has a dog,” and let F(x) be the statement “x has a ferret.” Express each of these statements in terms of C(x), D(x), F(x), quantifiers, and logical connectives. Let the domain consist of all students in your class. a. A student in your class has a cat, a dog, and a ferret. i. ∃x(C(x) ∧ D(x) ∧ F(x))
b. All students in your class have a cat, a dog, or a ferret. ii. ∀x(C(x) ∨ D(x) ∨ F(x))
c. Some student in your class has a cat and a ferret, but not a dog. iii. ∃x(C(x) ∧ F(x)∧¬D(x))
d. No student in your class has a cat, a dog, and a ferret. iv. ¬∃x(C(x) ∧ D(x) ∧ F(x))
e. For each of the three animals, cats, dogs, and ferrets, there is a student in your class who has this animal as a pet. v. (∃xC(x))∧(∃xD(x))∧(∃xF(x))
11. Let P(x) be the statement “x = x².” If the domain consists of the integers, what are these truth values? f. P(0)
vi. 0=0², True
vii. 1=1², True
viii. 2=2², False
ix. -1=(-1)², False
x. True, there exists a number that can be true for x=x² k. ∀xP(x)
14. Determine the truth value of each of these statements if the domain consists of all real numbers. a. ∃x(x³ = −1)
i. X=-1, True
b. ∃x(x4 < x²)
ii. X=.5, True
c. ∀x((-x)² = x²)
iii. True, - * - = +
d. ∀x(2x > x)
iv. False, Using negative numbers or 0 makes this statement false. 18. Suppose that the domain of the propositional function P(x) consists of the integer’s −2, −1, 0, 1, and 2. Write out each of these propositions using disjunctions, conjunctions, and negations. a. ∃xP(x)
ii. P(−2) ∧ P(−1) ∧ P(0) ∧ P(1) ∧ P(2)
22. For each of these statements find a domain for which the statement is true and a domain for which the statement is false. a. Everyone speaks Hindi.
i. True: Elected officials in Indian parliament
ii. False: People in United State
b. There is someone older than 21 years.
iii. True: people in United State
iv. False: kids in middle school.
c. Every two people have the same first name.
v. True: Group of boys with first name bob.
vi. False: Group of boys with first name bob and one bobs wife d. Someone knows more than two other people.
vii. True: A pregnant woman stuck on an island alone. (but she knows people from other places) viii. False: A child born on...
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