Fu Liu
Math 145
SOLUTIONS TO HOMEWORK 9
1. (10.4.5) Is there a bipartite graph with degrees 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 6, 6? (These can
be distributed in the two classes of nodes arbitrarily.)
Solution. No. Suppose there is a bipartite graph G on A t B

Fu Liu
Math 145
SOLUTIONS TO HOMEWORK 4
1. One hundred points are given inside a cube of side length one. Prove that there are four
of them that span a tetrahedron whose volume is at most 1/99.
(Hint. You might directly use the geometric fact that the vol

Fu Liu
Math 145
SOLUTIONS TO HOMEWORK 2
1. (3.6.3) Suppose n, m, k N. Prove the following identity in two ways:
n m
n
m
n
m
n m
n+m
+
+ +
+
=
.
0
k
1
k1
k1
1
k
0
k
(a) Use combinatorial interpretations.
(b) Write
(x + y)n+m = (x + y)n (x + y)m .
Apply

Fu Liu
Math 145
SOLUTIONS TO HOMEWORK 8
1. Let n 2. Suppose y = (y1 , y2 , . . . , yn2 ) is a sequence of length n 2 composed of
numbers in [n]. Recall that in class we describe an algorithm R which first construct a
matrix
x1 x2 xn2 xn1
,
y1 y2 yn2 yn1
a

Math 145
Fu Liu
Midterm
Solutions1
1
c
Copyright 2016
Fu Liu All Rights Reserved
February 10, 2016
1 (10 pts.)
Let n 2 be an integer. Recall that 2[n1] is the set of all subsets of [n 1].
[n]
We also denote by 2odd the set of all subsets of [n] of odd car

Fu Liu
Math 145
SOLUTIONS TO HOMEWORK 7
1. Prove that a graph G = (V, E) is a tree if and only if for any vertices u, v V , there
exists a unique path in G connecting u and v.
Proof. =: Suppose G is tree. Hence, G is connected and cycleless. Let u, v V
be

Fu Liu
Math 145
SOLUTIONS TO HOMEWORK 3
1. (2.3.1) See book for the problem.
Solution. Let S be the class of boys, A1 , A2 , A3 , and A4 the set of boys in the class that
like to play chess, play soccer, biking and hiking, respectively. Then the problem t

Fu Liu
Math 145
Homework 9
due on March 9, 2016
Sections of the textbook relevant to this assignment are: 10.110.4. (Materials covered by previous homeworks might also be used in this assignment.)
Recall that we say a bipartite graph G on A t B is a good

Fu Liu
Math 145
Homework 8
due on March 2, 2016
Sections of the textbook relevant to this assignment are: 8.3, 9.1, 9.2 and
10.1. (Materials covered by previous homeworks might also be used in this
assignment.)
Note. This homework has a few problems that

Fu Liu
Math 145
Homework 2
due on January 20, 2016
Sections of the textbook that are relevant to this assignment are: 3.53.6
and 2.1. (Materials covered by the last homework might also be used in this
assignment.)
In particular, make sure to read 2.1.12 a

Fu Liu
Math 145
Homework 3
due on January 27, 2016
Sections of the textbook relevant to this assignment are: 2.32.4. (Materials covered by previous homeworks might also be used in this assignment.)
1. (2.3.1) See book for the problem.
2. After finish Prob

Problem set 2 solution
1. (Putnam, 1978-A1) Let A be any set of 20 distinct integers chosen from the arithmetic
progression 1, 4, 7, , 100. Prove that there must be two distinct integers in A whose
sum is 104. [Actually, 20 can be replaced by 19.] (Junpin

Fu Liu
Math 145
SOLUTIONS TO HOMEWORK 6
1. Let G = (V, E) be a connected multigraph, and e E an edge of G. Let H = (V, E \ cfw_e)
be the graph obtained from G by removing the edge e. Suppose e = cfw_u, v. Prove that
the followings are equivalent:
(a) H ha

Fu Liu
Math 145
Solutions to Homework 1
n
n+1
1. (1.8.5) Prove that
+
= n2 ; give two proofs, one using the combinatorial
2
2
interpretation, and the other using the algebraic formula for the binomial coefficients.
Proof. Using combinatorial interpretat

Fu Liu
Math 145
Homework 6
due on February 17, 2016
Sections of the textbook relevant to this assignment are: 7.2 and 7.3.
(Materials covered by previous homeworks might also be used in this assignment.)
1. Let G = (V, E) be a connected multigraph, and e

Fu Liu
Math 145
Homework 4
due on February 3, 2016
Sections of the textbook relevant to this assignment are: 2.4, and 4.1
4.3. (Materials covered by previous homeworks might also be used in this
assignment.)
1. One hundred points are given inside a cube o

Fu Liu
Math 145
Homework 7
due on February 24, 2016
Sections of the textbook relevant to this assignment are: 8.1 8.3. (Materials covered by previous homeworks might also be used in this assignment.)
You might want to use results from Problem 1 of Homewor

Fu Liu
Math 145
INFORMATION FOR MIDTERM
The midterm will be based on Homeworks 15 and lecture notes up to Section 7.2 (including Section 7.2). Please see details below.
(1) The exam will consist of 56 problems.
(2) The difficulty level will be similar to

DEFINITIONS AND NOTATIONS
In this notes, we summarize definitions and notations that are either used frequenctly in
the course or hard to understand.
Definition. If x belongs to S, we say x is an element of S or x is in S. Write x S. If x is
not in S, we

Fu Liu
Math 145
SOLUTIONS TO HOMEWORK 10
1. There are 11 unlabeled graphs on 4 nodes. (See solution to Problem 4 in Homework 5).
Find the chromatic number of each of them. No justification is required.
Solution. See graphs below. The red number below each

Fu Liu
Math 145
SOLUTIONS TO HOMEWORK 5
1. (4.3.3) Define a sequence of integers In by I0 = 0, I1 = 1, and In+1 = 4In + In1 . Find a
formula for In .
Solution. The characteristic polynomial of the recurrence
relation
given by the problem
2
is x 4x 1, whic

Fu Liu
Math 145
Homework 1
due on January 13, 2016
Sections of the textbook that are relevant to this assignment are: 1.2, 1.3,
1.51.8, 3.13.4.
For all the homework problems (including this assignment and all other
assignments), unless otherwise specified

Fu Liu
Math 145
Homework 5
This homework will not be collected or graded.
Sections of the textbook relevant to this assignment are: 4.3, 7.1 and
7.2. (Materials covered by previous homeworks might also be used in this
assignment.)
1. (4.3.3) Define a sequ

Math 150A: Modern Algebra
Homework 1
This assignment is due at the end of class Thursday 8/4. As mentioned in class, your solutions
must be written in LaTeX. You do not necessarily have to use LaTeX for any images you include.
Complete exercise 2.2.3 from

Math 135B Midterm 2 Spring 2016
NAME(print in CAPITAL letters, ﬁrst name ﬁrst): “5%.-ij
NAME(sign): _
ID#: _
Instructions: There are four problems. Make sure that you have all 4 problems. You must show
all your work to receive full credit.
Points received