Homework Set 3 (Problems from Chapter 2) Problems from 2.1 2.1.1. Prove that a b (mod n) if and only if a and b leave the same remainder when divided by n. 2.1.2. If a Z, prove that a2 is not congruent to 2 modulo 4 or to 3 modulo 4. 2.1.3. If a, b are in
Solutions to Homework Set 3
(Solutions to Homework Problems from Chapter 2)
Problems from 2.1
2.1.1. Prove that a b (mod n) if and only if a and b leave the same remainder when divided by n.
Proof.
Suppose a b (mod n). Then, by denition, we have
a b = nk
MIDTERM 2 REVIEW QUESTIONS
MATH 3613
Studying the solutions to these questions will help you prepare for the
midterm exam. Many are (variations of) previously assigned homework questions. You should compare them to the previously assigned questions.
Quest
HOMEWORK 5
MATH 3613
Question 1. Compute the values of
34
mod 5 and 56
mod 7.
In each case give you answer as a nonnegative number less than the modulus
(we might call this a reduced solution). (Hint: Rather than gure out 56 and
the taking its remainder m
Math 3013 Homework Set 5
Problems from 2.1 (pgs. 134-136 of text): 1,3,11,12,13,16,23 Problems from 2.2 (pgs. 140-141 of text): 1,3,5,7,11,12 Problems from 2.3 (pgs. 152-154 of text): 1,2,3,4,5,7,13,15,19,29
Math 3013 Problem Set 5
Problems from 2.1
HOMEWORK 1
MATH 3613
1. Classical logic and proofs
Recall that an odd number is an integer of the form 2k + 1 for some k Z,
and an even number is an integer of the form 2k for some k Z.
Question 1. Prove that the sum of two odd numbers is an even number.
Math 3613 SOLUTIONS TO THE FIRST EXAM 9:30 10:20 , October 5, 2005 1. Denitions and Axioms (5 pts each) (a) What is the Well-Ordering Axiom for the set N of non-negative integers? Every non-empty subset of N has a least element.
(b). What precisely do we
Math 3613
Homework Problems from Chapter 4
4.1
4.1.1. Perform the indicated operations in Z6 [X ] and simply your answer.
(a) (3x4 + 2x3 4x2 + x 4) + (4x3 + x2 + 4x + 3)
(b) (x + 1)3
4.1.2. Which of the following subsets of R[x] are subrings of R[x]? Just
HOMEWORK 4 SOLUTIONS
MATH 3613
Question 1. The factorial of a natural number n is dened by
n! = n(n 1)(n 2) 3 2 1.
So 1! = 1, 2! = 2, 3! = 6, 4! = 24, etc.
(1) Factor 10! into a product of prime numbers.
Solution. Note that
1=1
2=2
3=3
4 = 22
5=5
6=23
7=7
Math 3013 Homework Set 6
Problems from 3.1 (pgs. 134-136 of text): 11,16,18 Problems from 3.2 (pgs. 140-141 of text): 4,8,12,23,25,26 1. (Problems 3.1.11 and 3.1.16 in text). Determine whether the given set is closed under the usual operations of ad
HOMEWORK 2
MATH 3613
Question 1. Let A, B be sets. Prove that
(A \ B) (B \ A) = (A B) \ (A B).
Proof. We need to prove that x (A\B)(B\A) x (AB)\(AB) in
order to see that these are the same sets. To see this, note that x A\B
x A x B, and in general, x S T
HOMEWORK 3
MATH 3613
Question 1. Let a, b, c Z with c = 0.
(1) Prove that if ac | bc, then a | b.
Proof. ac | bc means that bc = act for some t Z, but then bc act =
c(b at) = 0, but since c = 0, this is only possible if b at = 0, so
b = at and a|b.
(2) Pr
MIDTERM REVIEW QUESTIONS
MATH 3613
Studying the solutions to these questions will help you prepare for the
midterm exam. Many are (variations of) previously assigned homework questions. You should compare them to the previously assigned questions.
Questio
MIDTERM 2 REVIEW QUESTIONS
MATH 3613
Studying the solutions to these questions will help you prepare for the
midterm exam. Many are (variations of) previously assigned homework questions. You should compare them to the previously assigned questions.
Quest
LECTURE 3
Methods of Proof, contd
Last time we discussed some of the basic techniques for proving propositions. We began with the notion of
a direct proof and one of its implementations, the forward-backward method. Then we discussed an
alternative to the
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LECTURE 1
Logic and Proofs
The primary purpose of this course is to introduce you, most of whom are mathematics majors, to the most
fundamental skills of a mathematician; the ability to read, write, and understand proofs. This is a course
where proofs mat
Homework Set 2 (Homework Problems from Chapter 1) Problems from Section 1.1. 1.1.1. Let n be an integer. Prove that a and c leave the same remainder when divided by n if and only if a c = nk for some k Z. 1.1.2, Let a and c be integers with c = 0. Then th
MATH 3613
Homework Problems from Chapter 3
3.1
3.1.1. The following subsets of Z (with ordinary addition and multiplication) satisfy all but one of the
axioms for a ring. In each case, which axiom fails.
(a) The set S of odd integers.
(b) The set of nonne
Math 3613
Homework Problems from Chapter 4
4.1
4.1.1. Perform the indicated operations in Z6 [X] and simply your answer.
(a) (3x4 + 2x3 4x2 + x 4) + (4x3 + x2 + 4x + 3)
(b) (x + 1)3
4.1.2. Which of the following subsets of R[x] are subrings of R[x]? Justi
MATH 3613
Homework Set 1
1. Prove that not-Q not-P implies P Q
2. Prove that if m and n are even integers, then n + m is an even integer.
3. Prove that if n is an odd integer, then n2 is an odd integer.
4. Prove that if n is an integer and n2 is odd, then
MATH 3613
Homework Set 1
1. Prove that not-Q not-P implies P Q
2. Prove that if m and n are even integers, then n + m is an even integer.
3. Prove that if n is an odd integer, then n2 is an odd integer.
4. Prove that if n is an integer and n2 is odd, then
LECTURE 4
Methods of Proof, Contd
1. Review
Last week we discussed a variety of techniques for proving propositions of the form
PQ
.
Three of these are indicated diagrammatically below:
Direct Proof and the Forward Backward Method
P P1 P2 Q2 Q1 Q
Proof by
LECTURE 3
Methods of Proof, contd
Last time we discussed some of the basic techniques for proving propositions. We began with the notion of
a direct proof and one of its implementations, the forward-backward method. Then we discussed an
alternative to the
LECTURE 17
Polynomial Arithmetic and the Division Algorithm
Definition 17.1. Let R be any ring. A polynomial with coecients in R is an expression of the form
a0 + a1 x + a2 x2 + a3 x3 + + an xn
where each ai is an element of R. The ai are called the coeci
LECTURE 16
Homomorphisms and Isomorphisms of Rings
Having now seen a number of diverse examples of rings, it is appropriate at this point to see how two
dierent sets might be endowed with essentially the same ring structure.
Consider a set R consisting of
Math 3013 Homework Set 9
Problems from 5.1 (pgs. 300-301 of text): 2,4,6,8,10,12,17,19,23 Problems from 5.2 (pgs. 315-317 of text): 1,2,3,4,5,9,10,13 Problems from 7.1 (pgs. 394-395 of text): 7,8,9,10 1. (Problems 5.1.2, 5.1.4, 5.1.6, 5.1.8, 5.1.10,
Math 3013 Homework Set 7
Problems from 3.3 (pgs. 211-212 of text): 1,3,5,7,9,11 Problems from 3.4 (pgs. 226-229 of text): 1,5,7,16,20,26
1. (Problems 1,3,5,7,9,11 in 3.3) Find the coordinate vector of the given vector relative to the indicated order
Math 3013 Homework Set 2
Problems from 1.4 (pg. 68-69 of text): 1,3,5,9,12,13,15,25 1. (Problems 1.4.1, 1.4.3, and 1.4.5 in text). Reduce the following matrices to row-echelon form, and reduced row-echelon form. (a) 2 1 1 3 3 -1 (b) 0 -1 1 1 (c