CMPSCI 250: Introduction to Computation

David Mix Barrington and Hava Siegelmann

Fall, 2010

Homework Assignment #1

Posted Monday 13 September 2010

Due on paper in discussion class at 2:30 p.m., Tuesday 28 September 2010 (deadline moved from Thursday 23 September 2010 as of Wednesday 22 September)

There are nine questions for 80 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 three problems are taken from Rosen, Discrete Mathematics and its Applications, sixth edition, page 16-17. You don't need to buy that book, as the problems are copied here.

• Rosen #2 (5): Which of these statements are propositions? What is the truth value of each one that is a proposition?

  • (a) Do not pass go.

  • (b) What time is it?

  • (c) There are no black flies in Maine.

  • (d) 4 + x = 5.

  • (e) The moon is made of green cheese.

  • (f) 2ⁿ ≥ 100.

• Rosen #4 (5): Let p and q be the propositions "I bought a lottery ticket this week" and "I won the million dollar jackpot on Friday" respectively. Express each of these propositions as an English sentence:

  • (a) ¬p

  • (b) p ∨ q

  • (c) p → q

  • (d) p ∧ q

  • (e) p ↔ q

  • (f) ¬p → ¬q

  • (g) ¬p ∧ ¬q

  • (h) ¬p ∨ (p ∧ q)

• Rosen #10 (10): Let p, q, and r be the propositions "You get an A on the final exam", "You do every exercise in this book", and "You get an A in this class" respectively. Write each of these propositions using p, q, and r and logical connectives.

  • (a) You get an A in this class, but you do not do every exercise in the book.

  • (b) You get an A on the final, you do every exercise in this book, and you get an A in this class.

  • (c) To get an A in this class, it is necessary for you to get an A on the final.

  • (d) You get an A on the final, but you don't do every exercise in this book; nevertheless, you get an A in this class.

  • (e) Getting an A on the final and doing every exercise in this book is sufficient for getting an A in this class.

  • (f) You will get an A in this class if and only if you either do every exercise in this book or you get an A on the final.

• Problem 1.2.5 (10)

• Problem 1.5.3 parts e,f,g,h only (5)

• Problem 1.6.1 (15)

• Problem 1.7.6 (5) Note that the last sentence means "Explain why the given substitution did not lead to a valid tautology."

• Problem 1.8.1 (20)

• Problem 1.10.1 (5)

Last modified 27 September 2010