There will be in general one very short homework assignment due in each lecture, assigned just after the previous lecture, to be handed in on paper. Late papers will in general not be accepted. Students with major extenuating circumstances should contact the instructor.
Students are responsible for understanding and following the academic honesty policies indicated on this page.
Assignment 1, due Mon 9 September:
(1) In the Josephus Problem with 1024 people and a skip number of 2, who survives? Give a convincing argument that your claim is correct. You can see who survives the skip-2 game with 2, 4, 8, and 16 using Ensley-Crawley's app here .
(2) In the Grid Game explained in the page linked from here, give a convincing argument that if there are exactly three open squares, then the first player has a winning strategy.
Assignment 2, due Wed 11 September:
(1) Play the "Sequence Self-Test" game for Section 1.2, until you are ready to solve any such problem easily.
(2) Here is another recursively defined sequence. Start with any natural number. If the number is even, divide it by two to get the next number. If it is even, triple it and add 1. If we start with 1, we get 4, 2, 1, 4, 2, 1,... forever. Starting from 3 we get 10, 5, 16, 8, 4, 2, 1, 4, 2, 1,... forever. Try all the starting numbers up to 25. If you have time, try starting with 27.
Assignment 3, due Fri 13 September:
Let a1 and a2 be any two numbers in the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. (They may be equal to each other.) We can make a sequence like the Fibonacci sequence by using the rule "an+1 is the ones digit of an + an-1" or equivalently, using Java notation, the rule "an-1 = (an + an-1) % 10". For example, if we start with 6 and 8, the sequence goes 6, 8, 4, 2, 6, 8, 4, 2,... forever.
Your assignment is to give a general method to find the number ak for any positive integer k, given a1 and a2. For example, if a1 = 6 and a2 = 8, then the rule is this: "ak depends on the remainder when you divide k by 4. If the remainder is 1, the value is 6. If the remainder is 2, the value is 8. If the remainder is 3, the value is 4. And if the remainder is 0, the value is 2."
Start by trying a1 = 1 and a2 = 1, as in the normal Fibonacci series, and see if you can determine the long-term behavior.
(EC exercise 1.3.1 (c)) You come across three inhabitants of Smullyan's Island. A says "B and C are both lying". B says, "Only one of the other two is lying", and C says "At least one of us is lying". Which if any of them are telling the truth?
(EC exercise 1.3.21 (b)) Use truth tables to show that (p ∨ q) ∧ (¬p ∨ q) is equivalent to q.
(EC exercise 1.3.21 (e)) Use truth tables to show that (p ∨ q) ∧ (q ∨ r) is equivalent to (p ∧ r) ∨ q.
(EC exercise 1.4.14 (b)) Write the negation of this statement, simplifying so that no ¬ symbol is to the left of a quantifier. The type of the variables is "real number":
∃y:∀x: x + y = x.
(EC exercise 1.4.14 (d)) Write the negation as above, but now the type of x is "integer" and the type of y is "real number".
∀x:∃y: x/y = 2.
(EC exercise 1.4.15 (b)) Is the statement of 1.4.14 (b) true, or is its negation true?
(EC exercise 1.4.15 (d)) Is the statement of 1.4.14 (d) true, or is its negation true?
(EC exercise 1.4.16) Write the negation of each of the following statements as an English sentence. You might find it helpful to write a symbolic expression as an intermediate step:
(a) Every time you roll a "6", you have to take a card.
(b) There is a day in your life that is better than every other day.
(c) In every good book, there is a plot twist or a surprise ending.
(d) Every math course has a topic that everyone finds easy to do.
(EC exercise 1.5.17) Express each of the following statements using predicates and the quantifiers ∃ and ∀. Indicate the type f each of your variables. Useful predicates are M(x, y) "integer x is a multiple of integer y", E(n, d) "integer n ends in integer d", and Odd(x) "integer x is odd".
(a) If n is a multiple of 5, then n ends in 5 or n ends in 0.
(b) If n is not a multiple of 3, then n2 - 1 is a multiple of 3.
(c) For all odd integers a and b, there is no real number x such that x2 + ax + b = 0.
(d) For every real number y, if y ≥ 0, then there exists a real number x such that x2 = y.
(EC exercise 1.5.18) For each of the four statements in exercise 1.5.17 above, say whether it is true or false and explain your answer.
(EC exercise 1.5.30) Express each of the following using quantified statements over the domain S of college students and the predicates C(x) meaning "x is a computer science major" and D(x) meaning "x takes discrete mathematics". Which statements, if any, are equivalent to one another? Which, if any, are negations of one another?
(b) Some computer science majors take discrete mathematics.
(d) Not every computer science major takes discrete mathematics.
(e) All computer science majors do not take discrete mathematics.
(f) You must take discrete mathematics if you are a computer science major.
(g) Some computer science majors do not take discrete mathematics.
(EC Exercise 1.6.9) Use a truth table to decide whether each of the following argument structures is valid. If it is not, give an assignment of truth values to the propositional variables that makes each premise true and the conclusion false.
(b) premises p → (¬q ∧ r) and q, conclusion ¬p
(d) premises p ∨ (q ∧ r) and p → r, conclusion ¬q → r
(e) premises p ∧ (q ∨ r) and r → ¬p, conclusion r → q
(EC exercise 1.6.11 (b)) Use a truth table to show that the following argument is valid: "Steve votes for a Libertarian candidate if and only if both his wife Stella votes for a Democrat and his father Stan votes for a Republican. For Stella to vote for a Democrat, it is necessary that Stan not vote for a Republican. Therefore, Steve does not vote for a Libertarian."
(EC exercise 2.1.2) Decide which of the following you believe to be true or false. If the statement is false, give a specific counterexample. If you believe the statement is tre, provide the examples you tried as "evidence".
(a) For each n ≥ 1 and m ≥ 1, if m is odd and n is even, then m + n is divisible by 3.
(b) For each n ≥ 1, if n is odd and divisible by 3, then n2 - 1 is divisible by 8.
(c) For every n ≥ 2, if n is divisible by 3, then n(n+1)(n+2) is divisible by 4.
(d) For each n ≥ 1, if n is odd, then 4n - 1 is prime.
(e) For each n ≥ 1, if n is odd, then 2n - 1 is not divisible by 3.
(f) For each n ≥ 1 and m ≥ 1, if m and n are even, then mn - 1 is not a perfect square.
(g) For each n ≥ 1, if n is not prime, then neither is 2n - 1.
(h) For each n ≥ 1, if n is divisible by 4, then 3n + 1 is divisble by 4.
(EC exercise 2.1.8) Write a proof in the form of a letter from the AUTHOR to the READER of the statement "If n is odd, then 3n2 + 1 is divisible by 4".
(EC exercise 2.2.4) Provide a counterexample to each of the following statements about integers that is false. You do not need to prove the statements you believe to be true.
(a) If (a % c) = (b % c), then a = b.
(b) If (a % b) = c, then ((a + 1) % b) = c + 1.
(c) If a < b, then (a % c) + (b % c) = ((a + b) % c).
(d) If b < c, then (a % b) + (a % c) = (a % (b + c)).
(EC exercise 2.2.7) Prove each of the following propositions:
(b) If a divides b and a divides c, then a divides b - c.
(d) If c divides a and x is any nonzero integer, then cx divides ax.
(f) If 6 divides n3 - n, then 6 divides (n + 1)3 - (n + 1). (Hint: Multiply out (n + 1)3 - (n + 1) and simplify.)
(EC exercise 2.2.2)
Compute each of the following:
(b) 187 mod 11
(d) -24 mod 4 (Note: Here "mod" is the operator defined by EC, so that "m mod d" is always in the range from 0 through d - 1.)
(f) (9k2 + 5) mod 3
(EC exercise 2.2.15) Prove that if n is divisible by 3 and n is divisible by 4, then n is divisible by 12. (Hint: Write n = 12q + r and consider the cases.)
(EC exercise 2.2.27) If a and b are both odd integers, show that the polynomial x2 + ax + b cannot be factored. (Hint: They mean "factored over the integers", so they want you to show that it cannot be written as (x + c)(x + d) where c and d are both integers.)
(EC exercise 2.3.2, all parts)
(EC exercise 2.3.4, all parts) Remember that in each case you first (1) show that the value given for n = 1 satisfies the closed formula, and then (2) show that if the n-1'st term satisfies the closed formula, for any n, then so does the n'th term.
(EC exercise 2.3.9 part (b) only)
(EC exercise 2.3.13)
(EC exercise 1.5.4 parts (b), (d), and (f)) You may arrange the truth tables either their way or my way.
(EC exercise 1.5.28 part (b) only)
(EC exercise 2.2.21)
(EC exercise 2.3.5) (Use their hint.)
(EC exercise 2.2.14 part (b) only)
(EC exercise 2.2.19)
(EC exercise 2.3.12)
(EC exercise 2.4.1 part (b) only) [Hint: look at their solution for part (a)]
(EC exercise 2.4.2 part (b) only) [Same Hint]
(EC exercise 3.1.2)
(EC exercise 3.1.6)
(EC exercise 3.1.17, parts (a), (b), (d), and (f))
(EC exercise 3.1.27)
(EC exercise 3.1.32, parts (b), (d), and (e)
(EC Exercise 3.2.12)
(EC Exercise 3.2.14)
(EC Exercise 3.3.1 parts (b) and (d))
(EC Exercise 3.3.2 parts (b) and (d))
(EC Exercise 3.3.10)
(EC Exercise 3.3.12, parts (b) and (d) only)
(EC Exercise 4.1.5)
(EC Exercise 4.1.9, part (b) only)
(EC Exercise 4.1.11)
(EC Exercise 4.1.18)
(EC Exercise 4.2.2)
(EC Exercise 4.2.16)
(EC Exercise 4.3.3 parts (b) and (d) only)
(EC Exercise 4.3.20)
(EC Exercise 4.3.23)
(EC Exercise 4.4.6)
(EC Exercise 4.4.8)
(EC Exercise 4.4.18)
(EC Exercise 4.5.8, parts (b), (d), and (e) only)
(EC Exercise 4.5.14)
(EC Exercise 5.1.2, all parts)
(EC Exercise 5.1.4, parts (b), (d), and (f) only)
(EC Exercise 5.1.8, all parts)
(EC Exercise 5.1.12, parts (b), (d), and (f) only)
(EC Exercise 5.1.15, all parts)
(EC Exercise 5.1.21, all parts, read directions carefully)
(EC Exercise 5.2.2, all parts)
(EC Exercise 5.2.4)
(EC Exercise 5.2.7, all parts)
(EC Exercise 5.2.10, both parts)
Also be sure to review the practice test -- we will go over it in class and I will post results later on Wednesday.
(EC Exercise 5.3.4)
(EC Exercise 5.3.6)
(EC Exercise 5.3.9)
(EC Exercise 5.3.18)
(EC Exercise 5.3.26)
(EC Exercise 5.3.8)
(EC Exercise 5.3.15)
(EC Exercise 6.1.8)
(EC Exercise 6.1.13)
(EC Exercise 6.2.8)
(EC Exercise 6.2.10)
(EC Exercise 6.2.18) The answer to this question is easily available online -- the point is to explain it using the sum rules. Note that a hand with a pair cannot be a straight or a flush.
(EC Exercise 6.2.20) A "flush" is a hand where all cards are the same suit. Appendix A has all rules of games that are used in EC's examples and problems.
(EC Exercise 6.3.6)
(EC Exercise 6.3.8) Note that if there are two each, that is not "more heads than tails".
(EC Exercise 6.3.10)
(EC Exercise 6.3.15)
(EC Exercise 6.2.18) see above
(EC Exercise 6.2.20) see above
(EC Exercise 6.4.2)
(EC Exercise 6.4.7)
(EC Exercise 6.4.10)
(EC Exercise 6.4.24)
Read EC's Appendix B. We are going to be concerned with multiplying matrices, especially multiplying square matrices by themselves to form powers. Below are the matrices A, B, C, D, and E.
Compute AB, AC, BA, BC, CA, CB, DE, and ED.
Compute the i'th powers of each of these five matrices for all i from 0 through 5. The powers of D are harder to do by hand -- you may be able to find a matrix calculator on the web.
A = 0 1 B = 1 1 C = 1 0 D = 0.4 0.3 0.3 E = 2 0 1
1 1 0 1 1 1 0.1 0.2 0.7 1 1 0
0.6 0.3 0.1 0 2 1
Compute the powers of matrices A-E as assigned last time.
(EC Exercise 6.6.2)
(EC Exercise 6.6.4)
(EC Exercise 6.6.6)
(EC Exercise 6.6.8) Write down the ten results.
Last modified 25 November 2013