Problem Sets
for Math 103a, Winter 2012
Homework Rules: Every student must turn in their own
homework. You are encouraged to
work with other students on the problems, but you must write up the solutions
yourself. Do not go to the TA (or me)
and ask ÒHow do you do problem x?Ó
My answer will always be, ÒWhat did you try?Ó Be sure you understand all the
definitions involved in doing a problem before working on it (or asking about
it).
Unless
specified otherwise, page and problem numbers refer to the text by Gallian. Homework is to be turned in by midnight of the day
it is due.
Reading
for Week 1: Chapter 0
Problem
set 1. Due Tuesday, January
17. Turn in even-numbered problems
only!
pp. 21-23: 2, 4, 5, 6, 9, 15, 20, 21, 32, 34 (any method is
OK, but explain what you did).
Caution:
Future problem sets will be longer!
Solutions to
selected problems
Reading
for Week 2: Chapter 1 and pages 40-42 of Chapter 2.
Problem
set 2. Due Monday, January 23. Turn in even-numbered problems only!
Chapter
0, pp. 22-24: 16, 18, 20, 21, 24,
26, 36, 37, 54
Chapter
1, p. 35: 2, 3, 4, 5, 6
Chapter
2, p. 52: 1, 6, 8
Solutions to
selected problems
Reading
for Week 3: Chapter 2 and pp. 57-58 of Chapter 3.
Problem
set 3. Due Monday, January 30. Turn in even-numbered problems only!
Chapter
2, pp. 52-54: 5, 10, 12, 14, 18, 20, 21, 23, 24, 26, 28, 32, 34(a) (Not so easy;
use 32.)
Chapter
3, pp. 64-65: 1, 4, 9
Also:
Prove that if gcd(a,n) = 1 and b mod n = a mod n,
then gcd (b,n) =1.
See
Selected
solutions for problem set 2 for solution to Chapter 2, #5.
More solutions to
selected problems
Midterm
1 (on Monday, January 30 in class), will cover the
material in the first 3 homework
assignments. Most questions will be
similar to homework problems.
There
is no problem set due Monday February 6.
Problem set 4 will be posted on Wednesday Feb. 1 and will be due Monday,
February 13. (Yes, it will be
longer!)
Here
it is as promised!
Reading
for Week 4-5: Chapters 3 and 4.
Problem
set 4. Due Monday, February
13. Turn in even-numbered problems
only!
Chapter
3, pp. 64-69: 5, 7, 14, 18, 19, 20,
24, 26, 27, 28, 32, 42, 60
Chapter
4: pp. 81-86: 1, 8(b,c), 16,
22, 26, 34, 42, 44, 52
Solutions to
selected problems
Reading
for Week 6-7: Chapters 5, 6, and pp. 138-140 of Chapter 7.
Problem
set 5. Due Tuesday, February 21
(Monday is a holiday). Turn in
even-numbered problems only!
Chapter
5, pp. 113-117: 1(a,b,c), 2, 3(e,f),
6, 9, 13, 15, 18(b), 25, 26, 28, 29, 32, 43, 51
Chapter
6: pp. 133-135: 4, 5, 10, 14, 20, 32, 35
Also
(to be turned in):
A1: Find a 6-cycle in the symmetric group on
{1,2,3,4,5,6} whose square is the product (1,2,3)(4,5,6).
A2:
Prove that if gcd(a,n) = 1 and b mod n = a mod n,
then gcd (b,n) =1. Note:
this problem was assigned for Problem set 3. Some students did not understand that it
was to be turned in. If you omitted
it from your homework for 1/30 and therefore lost credit, please attach your
graded homework for 1/30 to have credit restored if the solution you turn in
now is correct.
Solutions to
selected problems
Reading
for Week 8-9: Rest of Chapter 7, Chapters 8&9
Problem
set 6 (Revised). Due Monday March
5. This is a long assignment. Get started early! Turn in even-numbered problems only!
Chapter
7, pp. 149-150: 1, 3, 6, 8, 22, 26,
28
Chapter
8: pp. 167-170: 8, 14 (not necessary to address general case), 16, 26, 34, 74
(Use 18 for R and 13 for M.)
Solutions to
selected problems (corrected)
Reading
for Week 9-10, Chapters 9&10
Problem
set 7. Due Monday March 12
Chapter
9, pp. 193-197: 4 (give reason), 10, 11, 13, 15, 16, 20, 24, 28, 30, 34, 44,
61, 70
Chapter
10: pp. 211-212: 4, 14
Solutions to
selected problems