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 intege
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 ass
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)
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 (
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
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 divi
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
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
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 s
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 the
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
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 yo
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,
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
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
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).
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 an
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 in