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

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 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

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

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

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

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 +

Problem Set Nine
Katherine Wilkinson
April 19, 2013.
Problem 1 (Ex. 3.13). Prove that 3x3 7y 3 +21z 3 = 2 has no integer solutions.
Solution. Any integer solution to 3x3 7y 3 + 21z 3 = 2 gives a solution to
[3x3 7y 3 + 21z 3 ]7
3
3
3
[3x ]7 + [7y ]7 + [21

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 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

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

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 Two Solutions
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 . We first check the base
case, n = 4. P (4) says that 5 24 34 , which is true since 5 24 = 80 and
34 = 81.
Now, sup

Problem Set One Solutions
Problem 1. Prove that if x Q, y
/ Q and x 6= 0 6= 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 6= 0, we can also write
x = c/d, where c Z+

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

Midterm 1 Topics
This midterm will test your knowledge of the material we have gone over in
class from the beginning of the semester until the end of Chapter II of the Course
Notes. That is, everything up to and including the proof of the Fundamental
Theo

Problem Set Four Solutions
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 a | c and b | c then ab | c.

Problem Set Five Solutions
Problem 1 (Ex. 1.5). Suppose b, c Z+ are relatively prime and a is a divisor
of b + c. Prove that
gcd(a, b) = 1 = gcd(a, c).
Solution. Well show that gcd(a, b) = 1. The proof that gcd(a, c) = 1 is
basically the same. Suppose k i

Problem Set Eight
Your Name Here
November 8, 2013.
Read the definition of partition (Def. 1.10) and the discussion that follows on
pg. 35 of the course notes.
Problem 1 (Ex. 1.18). Let f : X
Y be a function, and for every y Y
let Ay = f 1 (cfw_y). The s

Problem Set Seven
Your Name Here
November 1, 2013.
For all but the last the problem below, we assume that f is a function from
X to Y .
Problem 1 (Ex. 5.5). Suppose that f is surjective. Prove that every A X
satisfies
Y r f (A) f (X r A).
Show by example

Problem Set Three Solutions
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 are there from the
south-

Midterm 2 Topics
This midterm will cover all the material that we have gone over in class from
the following sections:
Ch 3.3: Operations on sets (plus Cartesian products),
Ch 3.4: Functions,
Ch 3.5: Image and preimage,
Ch 4.1: Equivalence relations 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

Conditional - a statement with a hypothesis and a
conclusion cfw_also known as a "if-rthen statement")
if then .
hypothesis H. H eonehisian
Hypothesis - the "if section of a conditional statement: it
is the independent mrioble [in other words it has to
ha

MATH 111
Homework 11
1. Determine which of the following functions are injective, surjective, or bijective. Prove your answer!
(a) f : R R where f (x) = 7x 5
2x2 1
x2 + 1
(c) f : Z Z Z Z where f (m, n) = (m + n, m n).
(b) f : R R where f (x) =
Solution :

PRIME, MODULAR
ARITHMETIC, AND
By: Tessa Xie
&
Meiyi Shi
OBJECTIVES
Examine Primes In Term Of Additive Properties & Modular
Arithmetic
To Prove There Are Infinitely Many Primes
To Prove There Are Infinitely Many Primes of The Form 4n+2
To Prove There

Section 1.3
1
Predicate Logic
Section 1.3:
1.3: Predicate Logic
Purpose of Section:
Section: To introduce predicate logic (or firstfirst-order logic)
logic which
the language of mathematics. We see how predicate logic extends the
language of sentential ca

1) Negate the following statement: given x (0, ), we have that x > y > 0 for some
y (0, ). [As usual, your answer should contain no nots.]
Solution. There exists x (0, ), such that for all y (0, ), x y or y 0.
2) Let A, B, C be subsets of U . [You can use