MAT 370, Spring 2014: Assignment #3
(Problems of the form Section ., #. are from your textbook.)
(1) Explain why (Un , +) is not a structure for any n N.
(2) Give an explicit isomorphism f : (U4 , ) (Z4 , +) where U4 = cfw_1, i, 1, i.
(3) Section 2, #1-4
LECTURE NOTES FOR MAT 370
DAMIEN PITMAN
0. Sets & Relations
A set has elements; write a S to denote "a is an element of S" and
a S otherwise.
/
Definition: Set-builder notation: cfw_ x | P( x ) where P is any property
Definition: For every set S and ever
Name:
MAT 370 - D. Pitman
Score:
/100
Exam 1 - March 5, 2014
Complete this exam without the aid of notes, books, electronic devices or other people.
The time limit for the exam is 75 minutes.
(1) (20 points) Let be dened on R by x y = |x y|. Then (R, )
LECTURE NOTES FOR MAT 370
DAMIEN PITMAN
Part IV: Rings and Fields
A ring is a set R with two binary operations + and (addition /
multiplication) satisfying
(R1) (R,+) is an Abelian group; (R2) Multiplication is associative;
(R3) Distributive laws (left a
LECTURE NOTES FOR MAT 370
PERMUTATIONS, COSETS, AND DIRECT PRODUCTS
DAMIEN PITMAN
8. Groups of Permutations
Definition: A permutation of a nite set A is a bijection A to A.
Definition: Let A = and denote by S A the set of all permutations on A.
Theorem: S
LECTURE NOTES FOR MAT 370
DAMIEN PITMAN
Part I: Groups ans Subgroups
1. Introduction & Examples
Definition: Let n N, n 2, then dene Zn = cfw_0, 1, . . . , n 1. Then, for any
x, y Zn , dene addition modulo (mod) n, denoted +n , by
x +n y =
x+y
x+yn
if x +
Name:
MAT 370 - D. Pitman
Score:
/100
Exam 2 - April 14, 2014
Complete this exam without the aid of notes, books, electronic devices or other people.
The time limit for the exam is 75 minutes.
(1) (4 points) Write Z100 in the canonical form from the Fun
MAT 370, Spring 2014: Assignment #1
(1) For the sets A = cfw_a, b and B = cfw_a, c, d, nd all of the following. List the elements in each set.
(a) A B
(b) A B
(c) A B
(d) P(A)
(2) Prove or disprove: the relation R given on the real numbers by xRy if and o
MAT 370, Spring 2014: Assignment #5
(The problems of the form Section ., #. are from your textbook.)
(1) (a) Section 5, #2
(b) Section 5, #6
(2) Section 5, #12
(3) Let G be an arbitrary group (written multiplicatively), and
Z(G) = cfw_x G | xy = yx for al
MAT 370, Spring 2014: Assignment #2
(1) Write the numbers below in the form z = |z|(cos + i sin ):
(a) 1 i
(b) 3 i
(2) Multiply (1 i)( 3 i) using
(a) Polar coordinates (use your answers from the previous problem)
(b) Cartesian coordinates
Verify that your
LECTURE NOTES FOR MAT 370
DAMIEN PITMAN
Part III: Homomorphisms and Factor Groups
13. Homomorphisms
def. homomorphism
Example. Sn Z2 (even->0)
Example. Z Zn (reduction mod n)
Properties of homs. 1) Image of e is e 2) phi(inverse)=inverse(phi)
3) phi(subgr
MAT 370, Spring 2014: Assignment #4
(The problems of the form Section ., #. are from your textbook.)
(1) (a) Section 3, #4
(b) Section 3, #6
(2) (a) Section 4, #4
(b) Section 4, #6
(3) (a) Use the Euclidean algorithm to write the gcd of 9 and 128 as a Z-l
STUDY GUIDE: MAT 370
PROFESSOR: DAMIEN PITMAN
TEXT: A FIRST COURSE IN ABSTRACT ALGEBRA
BY: FRALEIGH
Goals & Structure of Course
Content Goals. This course serves as an introduction to modern mathematics, where by modern I mean since the 18th century. Imag