HOMEWORK FOR DISCRETE MATHEMATICS STUDENTS

Math 2001: Discrete Mathematics, Fall 2008

Homework

Assignment	Assigned	Due	Problems
HW1	8/27/08	9/05/08	Assignment
HW2	9/3/08	9/10/08	Read Section 1.2. For practice, do problems 3, 5, 6 and 9 from Section 1.2. 1. Do problem 1.2.8. 2. Do problem 1.2.16. 3. You want to divide a piece of chocolate into 1"x1" square pieces to distribute to a group of children. The starting piece is an m"x n" rectangle. Show that it can be divided by breaking pieces of chocolate a total of mn-1 times, but not with fewer breaks. (Each "break" divides only one piece, and must be a straight line parallel to one of the sides of the piece.)
HW3	9/10/08	9/17/08	Read Section 1.6. 1. How many ways is it possible to climb a staircase if n steps if one is allowed to take either one or two steps at a time? 2. Show that the following 2 statements are true for all natural numbers m and n: (a) 0+m=m, (b) S(m)+n=S(m+n). 3. Show that m+n=n+m for every m and n.
HW4	9/19/08	9/24/08	Read Sections 1.1 and 3.1, along with the Sum Rule and Product Rule from Section 3.2. 1. Show that \|A\| is less or equal to \|P(A)\| for any set A. 2. By examining Pascal's triangle, make a conjecture about when C(n,k) is even. 3. Use counting arguments to show that C(2n,n) is between 2ⁿ and 4ⁿ.
Mid 1		9/26/08	No HW
HW5	10/2/08	10/8/08	Review pp. 90-93. 1. Do problem 3.1.20. 2. Do problem 3.1.22. 3. Let e_n be the number of equivalence relations on an n-element set. (a) Find e_0, e_1, e_2, e_3, e_4. (b) Show that e_n is between 2^n and 2^{n^2} if n>4. (Hints: for (a), it will be easier to count partitions. You might find it easier to count partitions by grouping them according to ``type'': 123/4, 124/3, 134/2, 234/1 are the four partitions of {1,2,3,4} of type abc/d. You don't have to write down all the partitions, just give the number of each type. For (b), find injections from P(n) to Eq(n) and from Eq(n) to P(n^2), where Eq(n) denotes the set of all equivalence relations on n.)
HW6	10/8/08	10/15/08	1. (a) Show that there are n! linear orderings of {1,2,...,n}. (b) Show that n! lies between 2^{n-1} and 2^{n^2}. 2. Let B(n) be the n-th Bell number (the number of equivalence relations on an n-element set). Show that the Bell numbers satisfy B(0) = 1; B(n+1) = Sum_{k=0 }^n C(n,k)B(k). (Hint: count equivalence relations on {0,1,...,n} by first choosing which elements are NOT related to n.) 3. Prove that if x, y and z are integers, then x(y+z) = xy + xz. You may use anything you know to be true about the arithmetic of natural numbers to prove this statement.
HW7	10/15/08	10/22/08	Read Sections 2.1-2.2. 1. Do problems 2.1.10 and 2.1.12. 2. Do problem 2.1.26, but use the value n=225 instead of n=45. 3. Do Advanced Exercise 1 in Section 2.1. (Note: The left hand side of the identity should be x^b - 1 instead of x^b.)
HW8	10/23/08	10/29/08	Read Sections 2.3-2.4. 1. Do problem 2.3.6. 2. Do problem 2.3.20. 3. Do Advanced Exercise 2.3.2.
Mid 2		10/31/08	No HW.
HW9	11/5/08	11/12/08	Read Sections 3.2 and 3.3. 1. (a) Use the product rule to prove that the number of functions from an n-element set to an m-element set is m^n. (b) Use a counting argument to show that the multinomial coefficient C(mn; n, n,..., n) is between m^n and m^(mn). (When m=2, this is the same problem as HW 4, Problem 3.) (Hint for (b): to show that C(mn; n, n,..., n) is bigger than m^n, describe a way to encode functions from n to m into an ordered partition of mn into m cells of size n.) 2. Use the Multinomial Theorem to find the coefficient of x^7y^8 in (2x^3 + 3x^2y + 5xy^2 + 7*y^3)^5. 3. Suppose that S is a subset of {1, 2,..., 2n} of size n+1. (a) Show that S must contain two integers a and b such that gcd(a,b)=1. (b) Show that S must contain two integers c and d such that c divides d. (Hint for (b): Label n pigeonholes with the odd integers 1, 3, ..., 2n-1, and let the elements of S be pigeons. Put pigeon k from S into the pigeonhole labelled with the largest odd divisor of k.)
HW10	11/14/08	11/19/08	1. Use inclusion/exclusion to determine how many numbers between 1 and 1000 are (a) not divisible by 2, 3, 5 or 7. (b) not perfect n-th powers for any n>1. (Not perfect squares, not perfect cubes, etc.) (c) square free. (That is, not divisible by n^2 for any n>1.) (Hint for (c): it is enough to consider elements not divisible by the square of any prime.) The next two problems concern the analogies between the binomial coeeficients and the Stirling numbers. 2. (a) Show that if p is prime and 0 < k < p, then p divides C(p,k). (b) Use (a) to prove that 2^p is congruent to 2 modulo p. 3. (a) Show that if p is prime and 1 < k < p, then p divides the Stirling number S(p,k). (b) Use (a) to prove that the Bell number B_p is congruent to 2 modulo p. (Hints for 2 and 3: Give a subset S = {i, j, ..., k} of {1,2,...,p}, define the ``rotation'' of S to be S' = {i+1, j+1, ..., k+1}, where these entries are considered modulo p. In 2, show that every subset S of size k lies in a p-cycle S, S', S'', ... under rotation. In 3, show that every partition P lies in a p-cycle P, P', P'', where P' is obtained from P by rotating each of its cells. Then argue that if the objects you are counting can be grouped into p-cycles, the number of them must be divisible by p.)
HW11	11/20/08	12/3/08	1. If you roll a fair die three times, what is the probability that the results are in nondecreasing order? 2. (a) How many positive integral solutions are there to x1+x2+x3+x4 +x5 < 50? (b) How many positive integral solutions are there to x1+x2+x3+x4 +x5+x6 = 50? (c) Which answer is bigger and why? 3. How many ways are there to arrange a deck of 52 cards with no adjacent hearts?
HW12	12/4/08	-	Be prepared to present the solutions to the following problems in class on 12/10/08. 1. Give a direct proof, a proof of the contrapositive, and a proof by contradiction of the following statement: if A, B and X are sets, (X intersect A) = (X intersect B), and (X union A) = (X union B), then A = B. 2. Write the first six axioms of set theory formally. (Consider the empty set axiom to be "There is a set with no elements". Consider the axiom of infinity to be "There is an inductive set".) 3. Write the negation of the formal statement of the axiom of union. 4. Write the Intermediate Value Theorem (from calculus) formally.