MATH 239 Assignment 1 - Suggested Solutions
1 (a) [3 marks] For nonnegative integers k, m, n, let A be the set of k-element subsets of {1, . . . , m + n}, so we have m+n . |A| = k But each subset A can be written uniquely as = , where is a subs
MATH 239 ASSIGNMENT 1
Due Friday, September 19, at NOON in drop boxes (at St. Jerome's for Sec 01, outside MC 4067 for Sec 02, 03, 04)
1 (a) Give a combinatorial proof of the identity m+n k where k, m, n are nonnegative integers. (b) Give an algebra
Page 1
Name:
MATH 239 QUIZ October 8, 2008
Surname: Signature: Id.#: D. Jao I. Goulden K. Purbhoo R. Christian LEC LEC LEC LEC 001 002 003 004
Question 1 2 3 4 Total
Value 4 6 6 4 20
Mark Awarded
Instructions: 1. Please check the box beside you
MATH 239 Quiz (Wednesday) - Suggested Solutions
1. We have ST (x) =
(, )ST
xw2 (, ) xw1 ()
(, )ST
= =
S T
xw1 () xw1 ()
S T
= =
1 |T |
xw1 ()
S
= S (x) |T | as required. 2. (a) Let Nodd denote the set of odd positive integers, and Neven de
MATH 239 Fall 2008 Assignment 8
Due Friday, November 28, noon. 1. The purpose of this problem is to prove that every non-maximum matching admits an augmenting path. Let M be a matching in a graph G. Let N be a matching which is larger than M . Let H
MATH 239 ASSIGNMENT 6
Due Friday, November 14, at NOON in drop boxes (at St. Jeromes for Sec 01, outside MC 4067 for Sec 02, 03, 04)
1. For each of the following graphs:
h g a b i f e d c h g f e i a b c d
G:
H:
(a) Find the breadth first search
MATH 239 Fall 2008 Assignment 4
Due Friday, October 17, noon. 1. (a) Let A = {1} {0}{0} {1} and B = {0} {1}{1} {0} . Are the elements of AB uniquely created? (b) Let A = {0}{0} {1}{0} and B = {1} {0}{1}{1} . Are the elements of AB uniquely created? 2
MATH 239 ASSIGNMENT 2
Due Friday, September 26, at NOON in drop boxes (at St. Jeromes for Sec 01, outside MC 4067 for Sec 02, 03, 04)
1. (a) Let k be a positive integer. Let S be the set of all binary strings with exactly k ones, where the weight of
Math 239 Introduction to Combinatorics Fall 2008
Note: No printed material will be distributed in class. All information about the course will be posted on the course web page: http:/www.student.math.uwaterloo.ca/math239 Sec. 1 2 3 4 Time 11:30MWF 12
MATH 239 ASSIGNMENT 3
Solutions
1. Let k, n be fixed non-negative integers. Show that the number of solutions to the equation t1 + . . . + tk = n, where t1 , . . . , tk {0, 2, 4, 6, 8}, is equal to [xn ] (1 - x10 )k , (1 - x2 )k
and determine this
MATH 239 Assignment 5 - Suggested Solutions
1(a) [1 mark] The vertices of An are given by the n-element subsets of {1, . . . , 2n + 1}, so the number of vertices in An is given by p= 2n + 1 , n n 1.
(b) [1 mark] For any vertex , the neighbours are
MATH 239 ASSIGNMENT 7
Suggested solutions
1. For each n 1, let Tn be the graph whose vertices are the {0, 1, 2}-strings of length n, in which two strings are adjacent if they differ in exactly one position. For example, in T2 , 00 is adjacent to 10
MATH 239 ASSIGNMENT 5 Due Friday, October 31, at NOON
1. Let An , n 1, be the graph whose vertices are given by the n-element subsets of {1, . . ., 2n + 1}, where two vertices are adjacent if and only if they are disjoint subsets. (For example, when
MATH 239 ASSIGNMENT 7
Due Friday, November 21, at NOON in drop boxes (at St. Jerome's for Sec 01, outside MC 4067 for Sec 02, 03, 04)
1. For each n 1, let Tn be the graph whose vertices are the {0, 1, 2}-strings of length n, in which two strings ar
Page 1
Name:
MATH 239 QUIZ October 7, 2008
Surname: Signature: Id.#: D. Jao I. Goulden K. Purbhoo R. Christian LEC LEC LEC LEC 001 002 003 004
Question 1 2 3 4 Total
Value 4 6 6 4 20
Mark Awarded
Instructions: 1. Please check the box beside you
MATH 239 Quiz (Tuesday) - Suggested Solutions
1. We have S (x) =
sS (2)
xw2 (s) xaw1 (s)+b
sS
= =
sS
xb (xa )w1 (s) (xa )w1 (s)
sS (1) S (xa ),
= xb = x as required.
b
2. (a) Let Nodd denote the set of odd positive integers, and Neven denote the
MATH 239 ASSIGNMENT 2 Solutions
1. (a) Let k be a positive integer. Let S be the set of all binary strings with exactly k ones, where the weight of a string is its length. Find the generating function for S, expressed as S (x) = an xn . (b) Use the b
MATH 239 Fall 2008 Assignment 4 Solutions
Total 20 marks 1. (a) (1 marks) Let A = {1} {0}{0} {1} and B = {0} {1}{1} {0} . Are the elements of AB uniquely created? Solution. No, the elements are not uniquely created. For example, the elements 0 A and
MATH 239 ASSIGNMENT 6 Solutions
1. For each of the following graphs:
h g a b i f e d c h g f e i a b c d
G:
H:
(a) Find the breadth first search tree, rooted at a, in which the vertices are added in alphabetical order whenever there is a choice. (
MATH 239 MIDTERM - Suggested Solutions
1(a) Let Ak be the set of k-tuples (c1 , . . . , ck ), with 1 ci 5 for each i = 1, . . . k. Define a weight function w on Ak , by w(c1 , . . . , ck ) = c1 + . . . + ck . Then an,k = [xn ]Ak .
k Now, Ak = N5 ,
MATH 239 ASSIGNMENT 3
Due Friday, October 3, at NOON in drop boxes (at St. Jerome's for Sec 01, outside MC 4067 for Sec 02, 03, 04) 1. Let k, n be fixed non-negative integers. Show that the number of solutions to the equation t1 + . . . + tk = n, whe