MAT 670
Abstract and Linear Algebra for Teachers
Spring 1999-2000
Problem Set 1

The Integers

p. 387: 1, 2, 3, 5, 6

p. 14: 2ab, 3, 5, 7c, 16, 17

p. 21: 1ace, 3

Rationals and Reals

1. For each of the following subsets of R, find the least upper bound (if it has one):

a. {1, 2, 3, 4}
b. {r Î Q ½ r2 < p }
c. {r Î R ½ r2 < p }
d. {r Î Z ½ r2 < p }
e. [0, 4]
f. (0, 4)
g. {1/n ½ n Î Z}
h. {1 - 1/n ½ n Î Z}
i. {n/(n + 1)½ n Î Z}
j. (- ¥, - 4)

2. Use the Intermediate Value Theorem to discuss the number of real roots of

x5 - 4x3 + 4x2 - 5x + 3. 3. If n and m are odd positive integers with n > m, give an example of a polynomial with real coefficients of degree n having exactly m real roots.

Complex Numbers

p. 395: 1, 2, 3, 4, 5, 6, 8

The Integers Modulo m and Modulo p

p. 40: 1, 3abc, 6