Introduction to Abstract Mathematics
Fall 2013
Assignment 4.3
Due September 27
Exercise 1. What amounts of money can be formed using only $2 and $5 bills? Be sure to
prove your answer is correct!
Exercise 2. Consider an instance of the stable marriage pro
Intro to Abstract Math
Spring 2009
Project 1
Due September 11
Go to the library and nd a book which covers the basics of set theory (and introductory
logic, if possible). Your assignment is to check this book out and bring it to class. The book
you choose
Intro to Abstract Math
Fall 2009
Homework 30
Due December 7
Exercise 87. Let a Q, a = 0. Prove that the function f : Q Q given by f (x) = ax is
an isomorphism. Conversely, show that if g : Q Q is an isomorphism then g (x) = ax for
some nonzero a Q.
Exerci
Intro to Abstract Math
Fall 2009
Homework 29
Due December 4
Exercise 84. Show that Q and R (under addition) are not cyclic. [Suggestion: Argue by
contradiction. If a were a generator, what would need to be true about a/2?]
Exercise 85. Let G, H and K be g
Intro to Abstract Math
Fall 2009
Homework 28
Due November 30
Exercise 81. Let G be a group and a G. Suppose |a| = n. Prove that if k N and k |n
then |ak | = n/k .
Exercise 82. Let G be a group, a G and |a| = n. Recall that for any k Z, ak =
agcd(k,n) . Us
Intro to Abstract Math
Fall 2009
Homework 27
Due November 23
Exercise 78. For each pair (a, b), nd gcd(a, b) and express it in the form ra + sb with
r, s Z.
a. a = 11, b = 3
b. a = 42, b = 77
c. a = 420, b = 288
Exercise 79. Let n N, n 2 and let a Zn . Pr
Intro to Abstract Math
Fall 2009
Homework 26
Due November 16
Exercise 75. Let G be a group with identity element e and let a G. Suppose that |a| = n.
a. Prove that ak = e if and only if n|k . [Hint: One implication follows from the laws of
exponents. For
Intro to Abstract Math
Fall 2009
Homework 25
Due November 13
Exercise 72. Let G be an abelian group and let a, b G. Prove that (ab)n = an bn for all
n N. [Hint: Use induction.]
Exercise 73. Let H = cfw_g : R R | g (x) = ax + b with a, b R and a = 0.
a. Pr
Intro to Abstract Math
Fall 2009
Homework 24
Due November 9
Exercise 69. Write out the Cayley tables for (Z2 , +2 ), (Z3 , +3 ) and (Z4 , +4 ).
Exercise 70. Let n N, n 2.
a. Let a Zn . Show that if x n a = x for all x Zn , then a = 1.
b. Show that (Zn , n
Introduction to Abstract Mathematics
Fall 2013
Assignment 4.2
Due September 27
Exercise 1. Given integers n and k , with 0 k n, the (n, k ) binomial coecient is
n
k
=
n!
,
k !(n k )!
where we dene 0! = 1.
a. Prove that if 1 k n 1, then
n1
n1
+
k1
k
=
n
.
Introduction to Abstract Mathematics
Fall 2013
Exercise 1. In this exercise we will provide another proof that
for the sake of contradiction that 2 is rational.
a. Let p, q N with p/q =
Assignment 5.1
Due October 4
2 is irrational. Assume
2. Show that 0 <
Introduction to Abstract Mathematics
Fall 2013
Assignment 1.1
Due September 6
Exercise 1. Consider Example 2 in the paper by Gale and Shapley that we discussed in
class today.
a. Explain why there are a total of 24 possible sets of marriages.
b. Choose an
Introduction to Abstract Mathematics
Fall 2013
Assignment 3.1
Due September 20
Exercise 1.Express the following statements symbolically, and determine if they are true
or false. You may assume that the universe of discourse is R.
a. For all x 1/4, there i
Introduction to Abstract Mathematics
Fall 2013
Assignment 2.2
Due September 13
Exercise 1. Use a proof by contradiction to show that if n = 2k + 1 for some integer k ,
then n is odd. [Note: While intuitively obvious, this fact does require some kind of pr
Introduction to Abstract Mathematics
Fall 2013
Assignment 2.1
Due September 13
Exercise 1. Let P and Q be statements. Verify the following logical equivalences by either
constructing a truth table or by using established equivalences.
a. P P P P P
=
=
b.
Introduction to Abstract Mathematics
Fall 2013
Assignment 1.2
Due September 6
Exercise 1. Determine the prime constituents of each of the following statements, and use
this to express these statements symbolically.
a. If I am tired or hungry, then I canno
Introduction to Abstract Mathematics
Fall 2013
Assignment 3.2
Due September 20
Exercise 1. Conjecture and prove a formula for the sum of the rst n Fibonacci numbers.
Exercise 2. For n 1, the nth harmonic number is dened to be
n
Hn =
k=1
Prove that for all
Introduction to Abstract Mathematics
Fall 2013
Assignment 4.1
Due September 27
Exercise 1. Let Fn denote the nth Fibonacci number (where, as in class, we set F0 = 0 and
F1 = 1). Prove that for all n 0,
n
n
1+ 5
1
1 5
.
Fn =
2
2
5
Exercise 2.[The Division
Introduction to Abstract Mathematics
Fall 2013
Assignment 5.3
Due October 4
Exercise 1. For each pair (a, b), nd gcd(a, b) as well as x and y so that xa + yb = gcd(a, b).
a. (14, 23)
b. (130, 150)
c. (34, 144)
Exercise 2. Let a, b, c Z. Prove that if a|c,
Introduction to Abstract Mathematics
Fall 2013
Assignment 5.2
Due October 4
Exercise 1. Let a, b, c Z. Prove that if c|a and c|b, then c|xa + yb for every x, y Z.
Exercise 2. Let m, n Z. Prove that if m|n and m|n + 1, then m = 1.
Exercise 3. Show that the
Intro to Abstract Math
Fall 2009
Homework 23
Due November 6
Exercise 66. Let (G, ) be a group and let a, b, c G. Prove the following statements.
a. Left cancellation: If a b = a c then b = c.
b. Right cancellation: If b a = c a then b = c.
c. The equation
Intro to Abstract Math
Fall 2009
Homework 22
Due November 4
Exercise 63. Prove or disprove the following statements.
a. Subtraction is a binary operation on Z.
b. Subtraction is a binary operation on N.
c. Division is a binary operation on N.
d. Division
Intro to Abstract Math
Fall 2009
Homework 21
Due November 2
Exercise 60. Prove that for all n N, f : In In is an injection if and only if it is a
surjection.
Exercise 61. Let and denote the elements of S3 with two-row representations
123
231
and
123
213
,
Intro to Abstract Math
Fall 2009
Homework 9
Due September 25
Exercise 25. Verify the following claims made in class.
a. For all n N and all 1 k n
n+1
k
=
n
n
+
.
k
k1
b. For all n, l N
l(l + 1) (l + (n 1) = n!
n+l1
.
l1
Exercise 26. Prove that for all n N
Intro to Abstract Math
Fall 2009
Homework 8
Due September 23
Exercise 25. Let a, b, c Z. Prove that if a|b and b|c then a|c.
Exercise 26. Prove that for all n N, 5n 1 is divisible by 4.
Exercise 27. Assume that n is an integer. Prove that for all n 2, n c
Intro to Abstract Math
Fall 2009
Homework 7
Due September 21
Exercise 22. Use induction to prove the power rule from Calculus. That is, for all n N,
dn
x = nxn1 .
dx
Exercise 23. Prove that for all n N
(1 + 2 + 3 + + n)2 = 13 + 23 + 33 + + n3 .
Exercise 2
Intro to Abstract Math
Fall 2009
Homework 6
Due September 18
Exercise 16. Negate the following statement in a meaningful way.
For every x Q there exists y N so that sin(xy ) = 1 or cos(x + y ) = 1/2.
Exercise 17. Consider the following statement: For ever
Intro to Abstract Math
Fall 2009
Homework 5
Due September 14
Exercise 13. Let x, y Z. Prove that if xy is odd then x and y are both odd.
Exercise 14. Prove that all prime numbers greater than 2 are odd.
Exercise 15. Show that log2 (3) is irrational.
Notat
Intro to Abstract Math
Fall 2009
Homework 4
Due September 11
Exercise 10. Let P , Q and R be statements. Verify the following logical equivalences.
a. P (Q R) (P Q) (P R)
=
b. (P ) (R (R) P
=
c. (P Q) P (Q)
=
d. P (Q R) (P Q) (P R)
=
e. P Q (P Q) (P ) (Q)