CMPSCI 250: Introduction to Computation

David Mix Barrington and Hava Siegelmann

Fall, 2010

Homework Assignment #2

Posted Thursday 23 September 2010

Due on paper in class, Thursday 7 October 2010

There are fourteen questions for 100 total points. Most are from the textbook, Mathematical Foundation for Computer Science. Note that the book has both Exercises and Problems -- make sure you are doing a Problem and not the Exercise with the same number. The number in parentheses following each problem is its individual point value.

Students are responsible for understanding and following the academic honesty policies indicated on this page.

• The first six problems are taken from Rosen, Discrete Mathematics and its Applications, sixth edition, pages as indicated. You don't need to buy that book, as the problems are copied here.

• Rosen #47-20 (5): Suppose that the domain (input variable type) of the propositional function P(x) consists of -5, -3, -1, 1, 3, and 5. Express these statements without using quantifiers,instead using only negations, disjunctions, and conjunctions (the operators ¬, ∨, and ∧):

  • (a) ∃x:P(x)

  • (b) ∀x:P(x)

  • (c) ∀x:((x ≠ 1) → P(x))

  • (d) ∃x:((x ≥ 0) ∧ P(x))

  • (e) ∃x:(¬P(x)) ∧ ∀x:((x < 0) → P(x))

• Rosen #48-32 (5): Express each of these statements using quantifiers. Then form the negation of each statement so that no negation operator is to the left of a quantifier. Next, express the negation in simple English. (Do not simply use the words "It is not the case that".)

  • (a) All dogs have fleas.

  • (b) There is a horse that can add.

  • (c) Every koala can climb.

  • (d) No monkey can speak French.

  • (e) There exists a pig that can swim and catch fish.

• Rosen #73-14 (10): For each of these arguments, explain which rules of inference are used for each step.

  • (a) Linda, a student in this class, owns a red convertible. Everyone who owns a red convertible has gotten at least one speeding ticket. Therefore, someone in this class has gotten an speeding ticket.

  • (b) Each of five roommates, Melissa, Aaron, Ralph, Veneesha, and Keeshawn, has taken a course in discrete mathematics. Every student who has taken a course in discrete mathematics can take a course in algorithms. Therefore, all five roommates can take a course in algorithms next year.

  • (c) All movies produced by John Sayles are wonderful. John Sayles produced a movie about coal miners. Therefore, there is a wonderful movie about coal miners.

  • (d) There is someone in this class who has been to France. Everyone who goes to France visits the Louvre. Therefore, someone in this class has visited the Louvre.

• Rosen #85-22 (5): Show that if you pick three socks from a drawer containing just blue socks and black socks, you must get either a pair of blue socks or a pair of black socks. Rosen #120-30 (5): Show that A × B ≠ B × A, when A and B are nonempty, unless A = B.

• Rosen #147-20 (5): Let f: ℜ → ℜ be a function from the real numbers to the real numbers, and let f(x) > 0 for all x ∈ ℜ. Show that f(x) is strictly decreasing if and only if the function g(x) = 1/f(x) is strictly increasing. [The definition of "h is strictly increasing" is ∀x: ∀y: ((x < y) → (h(x) < h(y))), and the definition of "strictly decreasing" is similar.]

• Problem 2.1.1 (5)

• Problem 2.3.1 (5)

• Problem 2.3.4 (5)

• Problem 2.6.1 (10)

• Problem 2.6.2 (10)

• Problem 2.6.4 parts b and c only (10)

• Problem 2.8.3 (10)

• Problem 2.11.2 (10) [Note that this is on the supplemental handout from Thursday 23 September, rather than in the printed volume.]

Last modified 22 September 2010