MAT 145 Homework #5 For Submission
This weeks problems concern the exponential generating function of the
Fibonacci numbers. We define the Fibonacci sequence by the recursion formula
F0 = 0
F1 = 1
F = F
n 2.
n
n1 + Fn2 ,
The generating function we are co
HW14
Math 108 SSII 2015
Read 4.6
Exercises
Due 8/27, 1pm at the class box.
3.3
3.4
4.1
4.2
4.3
4.4
4.5
4.6
#6e
# 2d,8
# 11f
# 4b
#8b
# 5b
# 10b,13
# 4a,5bh,10a
Instructors Solutions
3.3
6e The partition cfw_A, B of Z where A = cfw_x Z | x < 3 and B = Z A
HW17
Math 108 SSII 2015
Read 6.3
Exercises
Due 9/2, 1pm at the class box.
4.6 # 5j
6.3 # 1b,2b,4,6,19b
No solutions this time: Midterm grading took longer than hoped for.
Notable Student Questions
1. Why is it that in Z12 the class has no inverse?
3
2. If
HW19
Math 108 SSII 2015
Read 6.5
Exercises
Due Tuesday 9/8, 1pm at the class box.
(Monday, 9/7 is a holiday)
2.4 # 7d
3.4 # 2e
4.6 # 5l
6.1# 1h
6.2 #5
6.3 # 13
6.4 # 5b
6.5 # 2,5,10a,11
Make a list of the two most important theorems from each chapter we
HW20
Math 108 SSII 2015
Exercises
Due 9/9, 1pm at the class box.
6.4 # 6a,21b
6.5 # 10b,16a
I Proofs
On the nal exam you will be asked to give your favorite proof of one of the
following types.
direct
contraposition
contradiction
two part
bicondition
HW1
Math 108 SSII 2015
Read the Course Syllabus, Preface to Student, and 1.1
(All available on smartsite announcements.)
Exercises
Due Tuesday, 8/4, 1pm at the class box.
A video walk to the class box can be found in the smartsite announcements.
1.1
#1 b,
HW2
Math 108 SSII 2015
Read 1.2 then do the exercises from 1.2
Read 1.3 then do the exercises from 1.3
(Many students nd it helpful to read the exercises before reading the section
so they have some idea what to look for.)
Exercises
Due 8/5, 1pm at the cl
HW3
Math 108 SSII 2015
Read 1.4
Exercises
Due, 8/56, 1pm at the class box.
1.1
1.2
1.3
1.4
#4d,5h,12b
#13bd,
#9cd,10abcd,
#1ab,2b,5ab,9a,11ab
Note: These problems are from the 7th edition. The rst chapter of the 7th
edition is on smartsite under announcem
HW15
Math 108 SSII 2015
Read 6.1
Exercises
Due 8/31, 1pm at the class box.
NOTE: With this assignment we will have a zero tolerance policy regarding
the following rule:
Re-establish the context (so our reader knows what you are saying without
needing to r
HW12
Math 108 SSII 2015
Read 4.2 and 4.3
Exercises
Due 8/25, 1pm at the class box.
1.6
2.4
3.3
3.4
3.5
4.1
4.2
4.3
#1i
# 6k
#3c
# 2c
# 2e
# 1e,11c
# 1bj,3b,9b,10,16b
#3bd,4bd,13a
Instructors Solutions
1.6
1i Claim: For all odd integers m and n, if mn = 4k
MAT 145 Homework #6 For Submission
This weeks problems concern the number Tn of labeled trees consisting of
n vertices and n 1 edges, where n 1. Here, a tree is assumed to be always
connected.
Problem 1 (10 points). List all labeled trees with 4 vertices,
MAT 145 Homework #4 For Submission
Consider properly placed n pairs of parentheses. We denote by Cn the
number of such parentheses.
Examples:
C1 = 1, because ( ) is the only choice.
C2 = 2, because you have ( ) and ( )( ).
C3 = 5. They are: ( ), ( )( )
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 specied,
AUGUST 24, 2015
(1) Section 7.3, problem 3.
(2) Section 8.1, problems 1-3. (Be as rigorous as possible.)
(3) Section 8.5, problem 2 (a cut-edge is one whose removal disconnects G).
(4) Section 8.5, problems 3-4, 9-12.
(5) Show that if n > 2 then the compl
AUGUST 3, 2015
(1) A committee of nine people must elect a chairperson, a secretary, and a treasurer. In how many
ways can this be done?
(2) In a standard set of dominos, each domino may be represented by the symbol [x|y], where x and
y are (not necessari
AUGUST 10, 2015
(1) In Fall 2014, there were 35,415 students enrolled at UC Davis.
(a) Prove that there was some set of at least 52 students who all had the same rst initial and all
had the same last initial.
(b) Prove that there were at least two student
AUGUST 19, 2015
(1) Section 7.1, problems 1-7. For problems 1 and 2, assume that two graphs are the same if they are
isomorphic (i.e., if you can relabel the vertices of one graph to get the other graph).
(2) Section 7.2, problems 4-11.
(3) Section 7.3, p
1.
(a) How many ways are there to distribute 61 one-dollar bills to 20 people?
This is the same as ordering 61 stars and 19 dividers, or choosing 19 dividers out of 80 total
80
objects. There are
ways to do this.
19
(b) How many ways are there to distribu
1.
(a) How many distinct rearrangements of the word ORANGE are there?
We are ordering six distinct objects (letters), so there are 6! orderings.
(b) How many distinct rearrangements of the word BANANA are there?
One approach: If the six letters were disti
HW4
Math 108 SSII 2015
Read 1.5 and 1.6.
Most students want to read 1.7 to see more examples.
Exercises
Due 8/10, 1pm at the class box.
1.2 #4a
1.3 # 1ef,6bd
1.4 #5c,9b
1.5 # 1bdf, 2d, 3bg, 6ac, 7ab,12ab
1.6 #1b by construction, 1d by contradiction, 4a by
HW5
Math 108 SSII 2015
Read 2.1, 2.2, and 2.3
Exercises
Due 8/11, 1pm at the class box.
1.5
1.6
2.1
2.2
2.3
#2b,3c
# 1g,7f
# 1bd, 2abc,6b,7,13,19a,
# 1bd, 2i,10a,13a,19c,
#1FL,11ab
Note: These problems are from the 7th edition.
RULES for HW5: Put a box at
HW6
Math 108 SSII 2015
Read 2.4, and 2.5
Exercises
Due 8/12, 1pm at the class box.
1.2#12c
1.3 #10eg
1.5 #2e
1.6 #4c
2.1 # 6d
2.2 # 13c
2.3 #1j
2.4 # 3ab, 5ab,6bc,8b,13c
2.5# 6b,2,6a,13a
Instructors Solutions
1.2
12c A truth table for the propositional fo
Math 145, Winter 2012, Midterm 1
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*******************
NAME: printed
NAME: signature
STUDENT ID
***********************
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OLUTlOMB
1
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Problem 2 _l
Problem 3
1 Problem 4
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MAT 145 - Problem Set 1
Due beginning of lecture on September 30th
Problem numbers are from Discrete Mathematics, Lovasz, Pelikan, Vesztergombi. Collaboration is permitted; looking for solutions from external
sources (books, the web, material from previou