AMS301.2: Finite Mathematical Structures A, Spring 2011
Instructor: Estie Arkin, Math Tower 1-106, 632-8363, estie@ams.stonybrook.edu,
http:/www.ams.sunysb.edu/estie/estie.html;
Oce hours: Tentative: Tuesday 11-12, Wednesday 1-3 and by appointment. You ma
AMS 301
(Spring 2011)
Homework Set # 11 Solution Notes
Problem A: Let M be the math club and C the chess club, then N (M ) = 15, N (C ) = 12. Since
13 members belong to exactly one of the clubs, we know that N (M C ) N (M C ) = 13. Also,
N (M C ) = N (M )
AMS 301
(Spring 2011)
Homework Set # 10 Solution Notes
# 6, 7.1: (a). an = an1 + an2 . If the rst digit is a 0, there are an1 ways to complete the
sequence. If the rst digit is a 1, then the second digit must be a 0, and then there are n 2 digits
left, so
AMS 301.2
(Spring 2011)
Homework Set # 8 Solution Notes
# 10, 5.3: (a). We rst give each person a pear, and now distribute 5 identical apples and 3
identical pears to 3 distinct people:
5+31
5
3+31
3
(b). The apples can be distributed 35 ways, as each app
AMS 301.2
(Spring 2011)
Homework Set # 7 Solution Notes
# 14, 5.1:
Here we must select in which of the 6 places we should place the 2, and then in
which of the 5 remaining places we should place the 3. The 4 1s will be placed in the remaining
4 spots. So
AMS 301
Spring, 2011
Homework Set # 5 Solution Notes
Problem A: (a). How many internal nodes does the tree have? n = mi + 1, n = 81, m = 5, so
i = 16. n = i + l, l = 81 16 = 65.
(b). How many edges does the tree have? 81 1 = 80
(c). What is the height of
AMS 301
Spring, 2011
Homework Set # 4 Solution Notes
#1, 2.3: (j). The graph is bipartite, only 2 colours are needed.
(n). = 4. There is a wheel formed by nodes a,b,d,j,i,c, as in Example 2 so we need at least 4 colours.
Since the graph is planar, 4 colou
AMS 301
Spring, 2011
Homework Set # 3 Solution Notes
#2, 2.1: (a). In order for Kn to have an Euler cycle, every vertex must have even degree; thus, we must have
n 1 even, which means that n must be odd (and 3, to avoid the trivial case of K1 ).
(b). K2 h
AMS 301
Spring, 2011
Homework Set # 2 Solution Notes
# 4, Supplement II: A graph with n vertices can have at most n = n(n2 1) vertices; this would be the case
2
of a complete graph, Kn , in which all n vertices have degree (n 1). Thus, if we have 3 vertic
AMS 301
Spring, 2011
Homework Set # 1 Solution Notes
# 3, 1.1: (a) Nodes represent teams, and edges represent games between teams. The resulting graph is a
circuit on 5 nodes.
(b). Consider a graph as in part (a). It must have 5 nodes, each of degree 2. S
AMS 301 (Spring, 2011)
Estie Arkin
Homework Set # 9
Due in class on Thursday, April 28, 2011.
Recommended Reading: Tucker, 6.1, 6.2
2(b,c), 4(c), 14 Section 6.1
4, 20 Section 6.2
Homework Set # 10
Due in class on Thursday, May 5, 2011.
Recommended Reading
AMS 301.2 (Spring, 2011)
Estie Arkin
Homework Set # 7
Due in class on Thursday, March 31, 2011.
Recommended Reading: Tucker, 5.1, 5.2
Do Problems:
14, 18, 26 Section 5.1
8, 14, 22 Section 5.2
Homework Set # 8
Due in class at the beginning of class on Thur
AMS 301.2 (Spring, 2011)
Estie Arkin
Homework Set # 6
Due in class on Thursday, March 24, 2011.
Recommended Reading: 3.3 (read the introduction on page 113, and then the section Approximate Traveling
Salesperson Tour Construction, pages 117-120), 4.2, 4.1
AMS 301.2 (Spring, 2011)
Estie Arkin
Homework Set # 5
Due in class on Tuesday, March 15, 2011.
Recommended Reading: Tucker, 3.1, 3.2
Do Problems:
Problem A: Let T be a balanced 5-ary tree with 81 nodes.
(a). How many internal nodes does T have?
(b). How m
AMS 301.2 (Spring, 2011)
Estie Arkin
Finite Mathematical Structures A
Homework Set # 3
Due in class on Tuesday, February 22, 2011.
Recommended Reading: Tucker, 2.1, 2.2 (Chapter 2) Do Problems:
2 Section 2.1
For each of the graphs in problem #4(e,g) secti
AMS 301.2 (Spring 2011)
Estie Arkin
Finite Mathematical Structures A
Make sure to explain your answers! An answer such as yes or 20, even if correct, will get only partial
credit.
Homework Set # 1
Due in class on Tuesday, February 8, 2011.
Recommended Rea
AMS 301.2 (Spring, 2010)
Estie Arkin
Exam 3 Solution sketch
Mean 75.89, median 80, top quartile 87, high 100 (3 of them!), low 18.
1. (20 points) Build a generating function for ar in the following procedures. Remember to state
which coecient solves the i
AMS 301.3 (Fall, 2009)
Estie Arkin
Exam 3 Solution sketch
Mean 77.9, median 80, high 100 (4 of them!), low 13. 1. (20 points) Build a generating function for ar in the following procedures. Remember to state which coecient solves the initial problem. You
AMS 301.2 (Spring, 2010)
Estie Arkin
Finite Mathematical Structures A
Exam 3: Tuesday, May 11, 2010, 5:15-6:46pm
READ THESE INSTRUCTIONS CAREFULLY. Do not start the exam until told to do so. Make
certain that you have all 6 pages of the exam. You will be
AMS 301.3 (Fall, 2009)
Estie Arkin
Finite Mathematical Structures A
Exam 3: Monday, December 14, 2009, 5:15-6:46pm
READ THESE INSTRUCTIONS CAREFULLY. Do not start the exam until told to do so. Make certain that you have all 5 pages of the exam. You will b
AMS 301.2 (Spring, 2010)
Estie Arkin
Exam 2 Solution sketch
Mean 73.16, median 75, top quartile 82, high 95, low 36.
1. (5 points) Let T be a 4-ary tree with 200 internal nodes. How many leaves does the tree have?
(A correct guess with no work shown will
AMS 301.2 (Spring, 2010)
Estie Arkin
Finite Mathematical Structures A
Exam 2: Thursday, April 8, 2010
READ THESE INSTRUCTIONS CAREFULLY. Do not start the exam until told to do so. Make
certain that you have all 6 pages of the exam. You will be held respon
AMS 301.2 (Spring, 2010)
Estie Arkin
Exam 1 Solutions
Mean 77.97, median 82.5, top quartile 91, high 100 (2 of them), low 34.
1. (13 points) Are the two graphs shown below isomorphic? If so, give the isomorphism; if not,
give careful reasons for your answ
AMS 301.2 (Spring, 2010)
Estie Arkin
Finite Mathematical Structures A
Exam 1: Thursday, February 25, 2010
READ THESE INSTRUCTIONS CAREFULLY. Do not start the exam until told to do so. Make
certain that you have all 5 pages of the exam. You will be held re
Chapter 8: Inclusion Exclusion
Set notation:
AB
AB
AB
A
De Morgan rules:
AB =AB
AB =AB
Counting elements in sets:
N the number of elements in the universe.
N (A) the number of elements in the set A.
N (A) = N N (A).
N (A B ) = N (A) + N (B ) N (A B ).
N (
Section 7.1 Recurrence realtion models
Def: A Recurrence relation is a formula that counts the number of ways to do a procedure with n
objects based on the number of ways to do this procedure with fewer objects.
Examples:
an = 5an1 + 6an2
an = 3an1 + 2n2
Section 6.1 Generating Function models
Def: Suppose ar is the number of ways to select r objects in some procedure. Then g (x) is a generating
function for ar if
g (x) = a0 + a1 x + a2 x2 + ar xr +
Example:
(1 + x)n = 1 +
n
n2
nr
nn
x+
x +
x +
x
1
2
r