Problem Set 3
Due MONDAY Feb 10
Exercise 1. Prove that if n is a perfect square then n must have the form 4k or 4k + 1.
Hint: write n = m2 for some m Z+ . When m is divided by 4 there are four possible
remainders.
Solution. Write n = m2 , and use the divi

Your name here
Problem Set 4
Due Feb 14
Exercise 1. Suppose that z1 , z2 , w C. Prove that
(a) |z1 z2 | = |z1 | |z2 |,
(b) z1 z2 = z 1 z 2 ,
(c) ww = |w|2 .
Solution. Write z1 = x1 + iy1 and z2 = x2 + iy2 .
For (a) we compute
|z1 z2 | =
(x1 x2 y1 y2 )2 +

Solutions
Problem Set 1
Due Jan 24
Exercise 1. Prove there is no positive x Q such that x3 = 28.
Solution. To get a contradiction, suppose there is a positive x Q such that x3 = 28. If
we write x = a/b for positive integers a and b, then (a/b)3 = 28 can b

MT216
Problem Set 7
Due March 14
Let f : X Y be a function.
Exercise 1. Suppose that f is surjective. Prove that every A X satises
Y
f (A) f (X
A) .
Show by example that the claim is false if we omit the hypothesis that f is surjective.
Solution. Suppose

MT216
Problem Set 8
Due March 21
Exercise 1.
(a) Dene a relation on R R by setting (a, b) (c, d) if there is a nonzero real number
such that (a, b) = (c, d). Prove that is an equivalence relation.
(b) Let f : X Y be a function, and for every y Y let Ay =

Problem Set Ten
Katherine Wilkinson
April 26, 2013.
Problem 1. Prove that if 0R and 0R are additive identities of a ring R then
0R = 0R .
Solution. Assume that 0R and 0R are both additive identities of a ring R. We
want to show that if this is true, then

Problem Set Four
Katherine Wilkinson
February 15, 2013.
Problem 1 (Ex. 1.4). Suppose a, b, c Z. For each of the following claims,
give either a proof or a counterexample.
(a) If a | b and b | c then a | c.
(b) If a | bc then either a | b or a | c.
(c) If

MT216
Problem Set 5
Due Feb 21
Exercise 1. Prove that if z C is a root of unity then z = z 1 .
Solution. If z is a root of unity then z n = 1 for some positive integer n, and hence |z |n = 1.
As |z | R0 , this implies that |z | = 1. But we know also that

MT216
Problem Set 10
Due April 4
Exercise 1. Compute
(a) the order of [3] (Z/103Z)
(b) the order of [2] (Z/197Z) .
Solution. (a) As 103 is prime, it follows from the Little Fermat Theorem that [3102 ] = [1]
in (Z/103Z) , and so the order of three is a div

Math 2216 Introduction to Abstract Math
Homework 11:
Spring 2015
Due Wednesday April 22 (in class)
1. Chapter IV, Exercise 3.13.
Prove that 3x3 7y 3 + 21z 3 = 2 has no integer solutions.
2. Chapter IV, Exercise 3.14 (c,d,e,f).
Determine if each of the fol

Exam I
February 24, 2014
You have 50 minutes to solve any FIVE of the SIX problems below.
No books or notes are allowed.
Cheating will result in a failing grade for the course.
Good luck.
NAME
Problem 1. Find two integer solutions to 9x + 23y = 3, or expl

Solutions
Problem Set 2
Due Jan 31
Exercise 1. Dene b1 , b2 , . . . by b1 = 11, b2 = 21, and bn+1 = 3bn 2bn1 for n 2. Prove
that bn = 5 2n + 1 for every n Z+ .
Solution. Let P (n) be the statement bn = 5 2n + 1. We will use complete induction. The
stateme

Exam II
March 31, 2014
You have 50 minutes to solve any FIVE of the SIX problems below.
No books or notes are allowed.
Cheating will result in a failing grade for the course.
Good luck.
NAME
Problem 1. Find the smallest positive solution to
13x 9
(mod 33)

Problem Set One
Katherine Wilkinson
January 25, 2013
Problem 1. Prove that if x Q, y Q and x = 0 = y , then xy Q.
/
/
Solution. To get a contradiction, assume that xy Q. Then we can write
xy = a/b, where a Z and b Z+ . Since x Q and x = 0, we can also wri

MT216
Problem Set 9
Due March 28
Exercise 1. Consider the bijection
f : Z/150Z Z/3Z Z/50Z
dened by f ([z ]150 ) = ([z ]3 , [z ]50 ) Find the unique [z ]150 Z/150Z such that f ([z ]150 ) =
([2]3 , [48]50 ).
Solution. We need to solve the simultaneous congr

MT216
Problem Set 6
Due Feb 28
Exercise 1. Given sets A, B , and C , prove
(a) A
(B C ) = (A
B ) (A
C)
(b) A
(B C ) = (A
B ) (A
C ).
Solution. For part (a) we compute
xA
(B C ) x A and x B C
x A and x (B C )c
x A and x B c C c
x A and x B and x C
x A a

Math 2216 Introduction to Abstract Math
Homework 9:
Spring 2015
Due Wednesday April 8 (in class)
1. Chapter III, Exercise 5.8
Suppose A1 , A2 X. Prove or give a counterexample:
f (A1
A2 ) = f (A1 )
f (A2 ).
2. Chapter III, Exercise 5.9.
Suppose f 1 (f (A)

Math 2216 Introduction to Abstract Math
Homework 10:
Spring 2015
Due Wednesday April 15 (in class)
1. Chapter IV, Exercise 1.21
If a b (mod n), prove that gcd(a, n) = gcd(b, n).
2. Chapter IV, Exercise 2.3
Find all solutions to the pair of congruences
z 3

Math 2216 Exam 2 Solutions
Spring 2015
1. (20 points) Let z =
2
2
+
2
2 i
(a) Express z in polar form z = rei
(b) Compute z 85 and write it in the form a + bi.
(c) Compute
1
z
and z (in the form a + bi).
Solution. (a). Since sin = cos =
4
4
2
2 ,
we have

MATH2216
Homework 7 Solution
A
Typset the answers to Q1 and Q2 in L TEX.
1. Suppose that o(z) = 28, what is the order of z 16 ?
Answer: We want to nd the smallest positive integer d such that (z 16 )d = 1. That means
z 16d = 1. Since o(z) = 28, by Proposi

Mathematics 2216
Homework 8 Solution
1. Given sets A and B dene the symmetric dierence
A B = (A B)
(A B).
(1) Prove that A B = (A B c ) (B Ac ).
(2) Use the result from part 1 to prove that A B = B A.
Answer: 1)
A B = (A B)
(A B)
(from denition)
= (A B) (

Mathematics 2216
Homework 6
Due Friday 10/17
1. Suppose that a, b and c are positive integers, where a | c, b | c, and gcd(a, b) = 1. Prove
A
that ab | c. (Also, typset the answer to this question using L TEX)
Answer: By relative prime relationship, there

Section 8: Symmetric Groups
The point of this section is to establish some notation used in talking about the symmetric
groups Sn on finite sets cfw_1, 2, . . . , n. Of course, an element f of Sn is technically a set of ordered
pairs
cfw_(1, f (1), (2, f

Problem Set Two
Katherine Wilkinson
February 1, 2013.
Problem 1 (Ex. 2.19). Prove that 5 2n 3n for all integers n 4.
Solution. Let P(n) be the statement that 5 2n 3n . For the base case, lets
solve n = 4, P(4) :
5 24
34
5 16
81
80
81
Now for the induct

Problem Set Three
Katherine Wilkinson
February 8, 2013.
Problem 1. Suppose I build a jungle gym in the shape of a m n grid.
That is, the jungle gym measures feet from north-to-south, m feet from eastto-west, and n feet from bottom-to-top. How many paths a

Problem Set Six
Katherine Wilkinson
March 15, 2013.
Problem 1 (Ex. 3.11). For each of the following statements either provide a
proof or a counterexample.
(a) A
(A
B) = B
(b) A
(B
A) = A
B
(c) A (B
C ) = (A B )
(A C )
(d) A (B
C ) = (A B )
(A C )
(e) (A B