# CMPSCI 190DM: A Mathematical Foundation for Informatics

### Fall, 2014

This page contains the homework assignments for CMPSCI 190DM. In general there will be homework due at every lecture. Problems will be taken from the textbook by Ensley and Crawley, Discrete Mathematics: Mathematical Reasoning and Proof with Puzzles, Patterns, and Games, once all students have access to the book.

• Assignment #1 (Assigned 3 Sept, due in class 5 September):

E&C define a grid game played by two players on a four by four grid of squares. All squares are initially blank. On her move, a player chooses one row or one column and places X's in one or more blank squares in that row or column. (She may not choose a column that is all X's.) The winner of the game is the player who places the last X, that is, the first player who cannot move on her term loses.

A hard problem would be to determine who wins the four by four game if both players play optimally. Your assignment is to determine who wins the two by two game, and justify your conclusion by explaining how that player can win the game no matter what their opponent does. If you succeed in this, take a look at the three by three game.

• Assignment #2 (Assigned 5 Sept, due in class 8 Sept):

Section 1.2 of the book is about integer sequences such as (1, 3, 5, 7, ...). If we call this sequence A, then A(1) = 1 (the first element of the sequence), A(2) = 3, A(3) = 5, and so forth.

We can give a sequence by a list "(1, 3, 5, 7,...)" as I did above, or with a closed form like "A(n) = 2n - 1", or with a recursive definition like "A(1) = 1, and for any i > 0, A(i+1) = A(i) + 2". An equivalent form of the same recursive definition is "A(1) = 1, and for any i > 1, A(i) = A(i-1) + 2".

Here are your two exercises, #7 and #8 on page 21 of the book.

#7: For each of these sequences given in closed form, write out the first five (or more if necessary) terms of the sequence. Use your answer to discover a recursive formula for the sequence.

• (a) a(n) = 3n + 1
• (b) b(n) = 5n - 2
• (c) c(n) = 2n + 7
• (d) d(n) = 3n + 1
• (e) e(n) = 7n - 6
• (f) f(n) = 3n2 + n
• (g) g(n) = 22n-1 - 1

#8: For each of these sequences give recursively, write out the first five (or more if necessary) terms of the sequence. Use your answer to discover a closed formula for the sequence.

• (a) a(1) = 5, a(n) = 5a(n-1)
• (b) b(1) = 2, b(n) = 5b(n-1)
• (c) c(1) = 3, c(n) = 2a(n-1)
• (d) d(1) = 4, d(n) = 4 + d(n-1)
• (e) e(1) = 1, e(n) = 4 + e(n-1)
• (f) f(1) = 6, f(n) = 3 + f(n-1)

• Assignment #3, due 10 September:

(E&C 1.2.25, adapted) Last week I explained the Josephus game, which is best explained now by a web link to the authors' page. For a given number of people and a given skip number, you can follow the procedure and see who is the last person to survive. We say that J(p, s) number of that person. For example, playing the game as it comes up on the site, we can see that J(6, 2) = 5.

• (a) For each value of p from 2 through 12, determine J(p, 2).
• (b) If you did this correctly, there should be a pattern in the answers that you can figure out. Describe this pattern in words and use it to predict the answer for 13, 14, 15, and 16 people, still with a skip of 2.
• (c) Predict the answer for 31, 42, and 53 people. Note that the web site can't handle more than 30 people and just does it for 30 if you give it a higher number.

• Assignment #4, due 12 Sept: In the book, Section 1.3 exercise 11 parts b, c, and e only (since a and d are in the back of the book), exercise 21 parts b, d, and e, exercise 22.

• Assignment #5, due 15 Sept: In the book, Section 1.3 exercises 7 (parts b and d only) and 18, and Section 1.4 exercises 2, 9, and 16 (a and c only).

• Assignment #6, due 17 Sept: In the book, Section 1.4 exercises 19 and 20, and Section 1.5 exercises 1 (parts b, d, f, g only) and 4 (parts b, d, f only).

• Assignment #7, due 19 Sept: In the book, Section 1.5 exercises 17, 18, 28 (part b only), 30 (parts b, d, f, g only), and Section 1.6 exercises 5 and 7.

• Assignment #8 due 22 Sept:

E&C 1.3.9: Use the letter s to represent the statement "Chris likes to play soccer", r for "Chris likes to read", amd p for "Chris likes to eat pizza". Use these names along with the basic logic operations (AND, OR, and NOT) to write each of following English sentences in symbolic logic notation:

• (a) Chris likes pizza but he does not like soccer.
• (b) Chris likes to read and eat pizza, or he likes to play soccer.
• (c) Chris does not like to eat pizza, but he likes to play soccer or read.
• (d) Chris likes to do two of these things but not all three.

E&C 1.3.23 parts c and d only: Use the double negative property and DeMorgan's laws to rewrite each of the following as an equivalent statement that never has the NOT symbol ¬ outside of a parenthesized expression. The first one is done for you as an example. (The two DeMorgan laws say that ¬(x ∧ y) is equivalent to ¬x ∨ ¬y and that ¬(x ∨ y) is equivalent to ¬x ∧ ¬y. The Double Negative Law says that ¬(¬x) is equivalent to x.)

• (ex) ¬(p ∧ ¬q) is equivalent to ¬p ∨ ¬(¬q) by DeMorgan's law, and this is equivalent to ¬p ∨ q by the double negative property.
• (c) ¬(¬(¬p ∧ q))
• (d) ¬(p ∧ (q ∨ ¬ p))

E&C 1.6.7: Use truth tables to characterize each of the following propositions as a tautology (always true), a contradiction (always false), or neither:

• (p → (q ∧ r)) ∨ ((p ∧ q) → r)
• ((p → q) ∧ (q → ¬p)) → ¬p
• ((p → q) ∧ (¬p → r)) → (q ∨ r)

• Assignment #9, due 24 September: The worksheet "A Murder Mystery", sent by email.

• Assignment #10, due 26 September: The worksheet "Translating Predicates", handed out on paper. If possible, also look at the worksheet "Translating Quantifiers".

• Assignment #11, due 29 September: Complete the worksheet "Translating Quantifiers" if you can (definitely do parts 3 and 4). Also do the following translation exercises, with variables of type "dog", predicates T(x) for "dog x is a terrier" and S(x, y) for "dog x is sillier than dog y", and constant dogs c "Cardie" and d "Duncan". You will also need the "=" operator in symbolic statements.

1. "Every dog is sillier than some terrier who is siller than Cardie."

2. "There is a dog who is siller than Cardie but not sillier than any dog who is not Cardie or Duncan."

3. "For every dog, there is a different dog who is sillier than both the first dog and Duncan."

4. ∃x:∀y:∃z:S(x, y) → (T(z) ∧ S(y, z))

5. ∀x:∀y:∃z:S(x, y) → (¬T(z) ∧ S(x, z) ∧ S(z, y))

6. ∃x:∀y:(S(y, c) ∧ S(y, d)) → ((y = c) ∨ (y = d))

• Assignment #12, due in class 3 October 2014: First two more translations like those in Assignment 11:

• "Cardie is siller than every terrier except Duncan."

• ∀x:∃y:T(y)∧(∀z:S(x, z) → S(y, z))

Then in the book, do Exercises 2.1.3 and 2.1.10 on pages 96 and 97. Remember that exercises in blue are done in the back, so that you can get hints about 2.1.3 from the solutions to the similar 2.1.2, for example.

• Assignment #13, due in class 6 October 2014: First, two more English-to-symbols translations. The first has variables of type "dog" and predicates as in Assignments 11 and 12. The second has predicates of type "positive integer" and predicates E(x) for "x is even" and O(x) for "x is odd".

• "Every terrier is either sillier than some non-terrier or sillier than Cardie."
• "There is an odd number between any two different even numbers.

In the book, do Exercises 2.1.4 (parts b, d, f only), 2.1.12 (parts b, d, e only), 2.2.4, and 2.2.10.

• Assignment #14, due in class 8 October 2014: In addition to discussing Assignment #13 in class, we will look at the examples in Section 2.2 and the first example in Section 2.3. Write up Exercises 2.2.15 and 2.2.17.

• Assignment #15, due in class 10 October 2014: Along with 2.2.15 and 2.2.17, write up 2.2.14 (b) and 2.3.2.

• Assignment #16, due in class 14 October 2014: Exercises 2.3.3 (parts b, d, and f only), 2.3.4, and 2.3.5.

• Assignment #17, due in class 15 October 2014: Exercises 2.3.9b, 2.3.13, and the following: "Prove by induction that the sequence defined by the rules a1 = 1 and am = am-1 + m2 satisfies the closed form an = n(n+1)(2n+1)/6."

• Assignment #18, due in class Wed 22 October 2014: Exercises 2.5.2, 2.5.12, 2.5.14 parts b and c.

• Assignment #19, due in class Fri 24 October 2014: Exercises 3.1.2, 3.1.6, 3.1.12 (parts b, d, e), 3.1.17 (parts a, b, d, f, g), 3.1.28, 3.1.30.

• Assignment #20, due in class Mon 27 October 2014: Exercises 3.1.18, 3.1.25 (you are asked to find a formula for m(A ∪ B ∪ C ∪ D)), 3.2.10 (parts b and d), 3.2.20, 3.3.2 (parts b, d, e), and 3.3.10.

• Assignment #21, due in class Wed 29 October 2014: Do the remaining problems from last time (3.2.20, 3.3.2 bde, 3.3.10), plus 3.2.18, 3.3.6bc, and 3.3.13bc.

• Assignment #22, due in class Fri 31 October 2014: Exercises 4.1.1 (parts b and d), 4.1.5, 4.1.9, 4.1.11, 4.1.15 (this is a multiple choice question, not a question with parts -- explain your answer), 4.1.18.

• I forgot to post an assignment due Mon 3 November, sorry.

• Assignment #23, due in class Wed 5 November: Exercises 4.2.2, 4.2.7, 4.3.2 part (b) only, 4.3.4 parts (b) and (c), 4.3.12, 4.3.20, 4.3.22.

• Assignment #24, due in class Fri 7 November: Exercises 5.1.2, 5.1.4 (parts a, b, d, f), 5.1.8, 5.1.12 (parts a, b, d, f), 5.1.13 (parts b, d), 5.1.19.

• Assignment #25, due in class Mon 10 November: (Read Section 5.2, which we would have gone over in class today if either of you were there.) Exercises 5.2.2, 5.2.4, 5.2.10, 5.2.15.

• Assignment #26, due in class Fri 14 November: (Read Section 5.3) Exercises 5.3.2 (b, d), 5.3.4, 5.3.9, 5.3.18, 5.3.20, 5.3.22

• Assignment #27, due in class Mon 17 November: (Read Section 5.4) Exercises 5.4.2, 5.4.4, 5.4.8, 5.4.12, 5.4.18, 5.4.20.

• Assignment #28, due in class Wed 19 November: (Read Sections 6.1, 6.2) Exercises 6.1.2 (b, c), 6.1.6, 6.1.10, 6.1.20, 6.2.2 (b, c), 6.2.6.

• Assignment #29, due in class Fri 21 November: (Read Section 6.3) Exercises 6.2.14, 6.2.18, 6.3.2, 6.3.6, 6.3.10, 6.3.15.

• Assignment #30, due in class Mon 24 November: (Read Section 6.4) Exercises 6.3.19 (parts a, b), 6.4.2, 6.4.4, 6.4.7, 6.4.10, 6.4.20.

• Assignment #31, due in class Mon 1 December: (Read Appendix B) Review the exercises for Appendix B, which have solutions in the text. Then consider the matrix

``````
1/2  1/4  1/4
A  =      1/4  1/4  1/2
1/2  1/4  1/4
``````

and compute the matrices A^2 (meaning "A times A"), A^3, A^4, and A^{10}.

• Assignment #32, due in class Tue 3 December: (Read Section 6.6, pp. 482-486) Exercises 6.6.2, 6.6.4, 6.6.6, 6.6.8, 6.6.10. If you do not have the appropriate dice available, you can simulate them using the random integer generator here.