Assignment--1

NQ 1. Let P(x,y) be the statement “house x has passed inspection y”, where the

universe of discourse for x consists of all the houses being constructed in a new subdivision and for y consists of all the inspections required by the locality, e.g., zoning, electrical, and occupancy. Translate each of the following statements into English. a. ∃x∀y P(x,y) b. ∃x∃y P(x,y) c. ∀x∃y P(x,y) d. ∀x∀y P(x,y) e. ∀y∃x P(x,y) f. ∃y∀x P(x,y) g. ∃y∃x P(x,y)

NQ2. In question 1, which two statements are equivalent? NQ3. For each statement in question 1, first negate the statement, then

transform the statement so that no negation symbol is to the left of a quantifier, and finally translate the resulting statement as directly as possible into English.

NQ4. Let K(x,y) be the statement “x knows y”, where the universe of discourse for both x and y is everyone in town. Translate the following statements into symbolic form using quantifiers and the predicate K. a. Janice knows everyone. b. Everyone is known by someone. c. Everyone knows Mr. Bainbridge. d. No one knows himself (or herself). e. Everyone who knows Amy knows John. f. Everyone knows someone. g. No one knows everyone. h. No one known by Amir is also known by Sara. i. There is someone who knows no one besides himself (or herself). j. Janet knows exactly one person besides herself. k. Mr. Ting is known by exactly one person. l. There is exactly one person that everyone knows.

NQ5. Given the propositional function P(x,y) where the universe of discourse for both x and y is {a,b,c}, use disjunctions and conjunctions to express the following propositions without the use of quantifiers. a. ∃x∃y P(x,y) b. ∃x∀y P(x,y) c. ∀x∀y P(x,y)

d. ∀x∃y P(x,y) NQ6. Let C(x) be the statement “x is a composite number” (composite means not

prime) and D(x,y) be the statement x and y have a common divisor (larger than 1), where the universe of discourse for both x and y is the set of positive integers. Translate the following statements into symbolic form using quantifiers and the predicates C and D. a. There are two (distinct) integers that have a common divisor. b. If x and y are both composite, then they have a common divisor. c. Exactly two positive integers less than 5 have a common divisor. d. If x is composite, then there is a positive integer less than x that

shares a divisor with x. e. If two distinct integers have a common divisor then they are both

composite. NQ7. Let C(x) be the statement “x had a camera” and P(x,y) be the statement “x

took a picture of y”, where the universe of discourse for both x and y is the people at Sudhir’s party last night. Translate the following statements into symbolic form using quantifiers and the predicates C and P. a. No one at the party took a picture of Mani. b. Mani did not have a camera. d. Sara did not take a picture of Helen. d. Ama took a picture of everyone except Mani. e. No one at the party did not have a camera. f. Exactly one person at the party did not have a camera. g. There was a person at the party who had a camera and another

person who did not have a camera. h. Exactly two people at the party had a camera. i. Exactly one person at the party had a camera but did not take any

pictures. j. Everyone at the party who had a camera took a picture of at least

one person. k. There was a person at the party who did not have a camera but who

took a picture of Helen. l. There were at least two people at the party who took a picture of the

same person. m. There were a pair of people at the party such that everyone at the

party had their picture taken by at least one of the pair. n. Every person at the party took a picture of another person at the

party and had his or her picture taken by a third person. NQ8. Translate the following statements into symbolic form. Define your

predicates and the relevant universe of discourse. a. Two positive integers are relatively prime if and only if neither of

them is divisible by any of the prime factors of the other. b. There is an integer that can be expressed as the sum of two squares

of integers in two different ways, i.e., the number is the sum of the squares of two distinct integers and the sum of the squares of two

other distinct integers. (Consider using a predicate over four variables.)

NQ9. Let P(x,y) be the statement x2 – y2 > 0, where the universe of discourse for x and y is the set of all integers. What is the truth value of each of the following statements? a. P(5,4) b. P(1,1) c. P(-3,-1) d. ∀x∀y (P(x,y) ∨ (x ≤ y)) e. ∃y∀x P(x,y) f. ∀x∃y P(x,y) g. ∀y∃x P(x,y) h. ∃y∀x (P(x,y) ∨ x = 0) i. ∃x∀y (P(x,y) ∨ x = 0)

NQ10. In each of the following statements assume the universe of discourse is the set of real numbers. What is the truth value of each statement? Explain. a. ∃x∀y (x2 + y2 ≥ 0) b. ∃x∃y ((x < y) → (x2 < y2)) c. ∀ x ∃ y (x + y > y + x) d. ∃ x∀y (y ≠ 0 → xy = 1) e. ∀x∀y (x2 < 2y) f. ∀x∃y (x + y = 5) g. ∀x∃y (y2 = x) h. ∃y∀x (x2 < y + 1) i. ∃x∀y (x + y = y) j. ∃x∃y (x + y = 0 ∧ x + 2y = 1) k. ∀x∃y∃z (x = y + z)

NQ11. Transform each statement into an equivalent statement in which negation operators appear only immediately before predicates. a. ∀x¬∀y P(x,y) b. ¬∃x∀y∀z P(x,y,z) c. ¬∀x¬∃y¬∀z ¬P(x,y,z) c. ¬ ∃x (∃y P(x,y) ∨ ∀y Q(x,y)) d. ¬∀x (P(x) ∧ ∃y Q(x,y)) e. ¬∃x ( P(x) ∨ ¬∀y ( P(y) ∧ Q(x,y) ) ) f. ¬∀x ( ∃y∀z P(y,z) ∧ ¬∀y∃z (Q(x,y,z) )

NQ12. Use quantifiers to express the conditions under which x is the largest integer in a set S of integers.

NQ13. Use quantifiers to express the conditions under which G(x,y) is true, where G(x,y) is the statement x2 > y2 and the universe of discourse for both x and y is the set of integers.

NQ14. Use quantifiers to express the following propositions a. x raised to the power of a raised to the power of b equals x raised to

the power of (a times b) b. Goldbach’s conjecture: Every integer greater than 2 is the sum of

two primes. c. Fermat’s last theorem: “… it is impossible to write a cube as the sum

of two cubes, a fourth power as the sum of two fourth powers and in general any power beyond the second as the sum of two similar powers”.

d. The product of any three consecutive integers is divisible by 6. e. The number of primes is infinite.