Math 300.1Fall 2013
Additional problems due September 24
1. Prove that, for all real numbers p 1 and all natural numbers n, (1 +
p)n 1 + np. (Hint: Use induction on n.)
2. Prove that, for every integer n 5, 2n > n2 .
3. Prove that, for each integer n 12,
Math 300.1 Fall 2013
Solutions for Assignment Due Thursday, October 10
11. Prove that gcd(ad, bd) = |d| gcd(a, b)
If d = 0, this says gcd(0, 0) = 0. But we know that this is
correct. So we can now assume d = 0.
Theorem 2.24 says that if d is a nonnegative
Math 300.1 Fall 2013
Solutions for Assignment Due Thursday, November 21
In problems 14, you will construct the integers from the set of whole numbers, W.
1. Let X = W W. Dene a relation on X by
(m, n) (p, q ) if and only if m + q = p + n.
Show that is an
Math 300.1 Fall 2013
Solutions for Last Assignment
1. Let X and Y be sets and let I (X, Y ) be the set
I (X, Y ) = cfw_f F (X, Y ) : f is injective .
(1)
Prove that, for n m,
|I (Nn , Nm )| =
m!
= m (m 1) (m n + 1).
(m n)!
(2)
There are several ways to th
Math 300.1 Fall 2013
Solutions for Assignment Due Thursday, November 14
28. Construct addition and multiplication tables for Z3 and nd, if possible,
multiplicative inverses of each of the elements in the set.
+
[0]
[1]
[2]
[0]
[0]
[1]
[2]
[1]
[1]
[2]
[0]
Math 300.1 Fall 2013
Solutions for Assignment Due Thursday, October 24
101. Prove or give a counterexample: lcm(gcd(a, b), gcd(a, c) = gcd(a, lcm(b, c).
Let p1 , . . . , pn be the distinct primes dividing at least one of a, b, c. Then
we can write
n
n
pxi
Math 300.1 Fall 2013
Solutions for Assignment Due Thursday, October 17
83. Let a, b, c be nonzero integers. Their greatest common divisor gcd(a, b, c) is
the largest positive integer that divides all of them. Prove that gcd(a, b, c) =
gcd(a, gcd(b, c).
Le
Math 300.1 Fall 2013
Solutions for Assignment Due Thursday, September 26
From Chapter 4
11. Prove by induction that, for all natural numbers n,
12 + 22 + 32 + + n2 =
n(n + 1)(2n + 1)
.
6
The base case is n = 1. The left hand side of the equation
is 12 = 1
NOTES FOR MATH 300 FALL 2013
CARDINALITY
GEORGE AVRUNIN
These notes are a complement to Chapter 6 of the textbook. They
are in preliminary form and intended only for use by students in Math
300 this semester.
1. Finite Sets
1.1. Cardinality and Finite Set
Math 300.1 Fall 2013
Solutions for Assignment Due Thursday, October 3
1. Show that, for ever integer n 0, that the sum of the binomial coecients
is 2n :
n
n
n
n
+
+
+ +
= 2n .
0
1
2
n
The binomial theorem gives
n
2n = (1 + 1)n =
i=0
n ni i
1 1=
i
n
i=0
n
Math 300.1 Fall 2013
Solutions for Assignment Due Tuesday, September 17
For 16, Determine which of the following sentences are statements. What
are the truth values of those that are statements?
1.1 7 > 5
True statement.
1.2 5 > 7
False statement.
1.3 Is
Math 300.1 Fall 2013
Review Sheet for Final Exam
Monday, December 9 10:3012:30 in LGRT 121
General Information: If, due to an emergency, you are unable to take the
exam at the scheduled time, it is your responsibility to notify me at the earliest
possible
Math 300.1 Fall 2013
Review Sheet for Midterm Exam
Thursday, October 31
General Information: The exam will be given in class on Thursday. If, due
to an emergency, you are unable to take the exam then, it is your responsibility
to notify me at the earliest
Math 300 Midterm Solutions
October 31, 2013
1. Give precise and complete denitions of the following terms:
(a) (3 points) Two statements are logically equivalent if and only if . . .
Solution: each of them implies the other. (So one is true if and only if
Math 300.1 Fall 2013
Assignment Due Tuesday, September 17
For 16, Determine which of the following sentences are statements. What
are the truth values of those that are statements?
1.1 7 > 5
1.2 5 > 7
1.3 Is 5 > 7?
1.4 2 is an integer
1.5 Show that 2 is n
Math 300.1 Fall 2013
Last Assignment
1. Let X and Y be sets and let I (X, Y ) be the set
I (X, Y ) = cfw_f F (X, Y ) : f is injective .
(1)
Prove that, for n m,
|I ( n , m )| =
m!
= m (m 1) (m n + 1).
(m n)!
(2)
2. Given that N N is countable (as proved i
Math 300.1 Fall 2013
Assignment Due Thursday, November 21
In problems 14, you will construct the integers from the set of whole numbers, W.
1. Let X = W W. Dene a relation on X by
(m, n) (p, q ) if and only if m + q = p + n.
Show that is an equivalence re
Math 300.1 Fall 2013
Assignment Due Thursday, October 3
1. Show that, for ever integer n 0, that the sum of the binomial coecients
is 2n :
n
n
n
n
+
+
+ +
= 2n .
0
1
2
n
2. (a) Show that, for every integer n 0, that the alternating sum of the
binomial coe
M300.1Fundamental Concepts of Mathematics
Fall 2013 Syllabus
Instructor:
Oce:
Telephone:
E-mail:
Required textbook:
Web page:
Oce hours:
George Avrunin
1335D LGRT
545-4251
avrunin@math.umass.edu
William Gilbert and Scott Vanstone, An Introduction to Mathe