MATH 09829 MATH 122B UCSB

MODEL ANSWERS TO THE SIXTH HOMEWORK 1. Chapter 3, Section 6: 1 There are two cosets. The first coset is [1] = N, the second is the coset containing -1, which is the set of all negative real numbers. [1] [1] = [1], [1] [-1] = [-1] [1] = [-1] and [-

MODEL ANSWERS TO THE FIRST HOMEWORK 4. (a) Note that x (A B) x (A B) / either x A or x B / / x A B , either x A or x B and the result follows. (b) Similar to (a). 6. There are many ways to prove this. Perhaps the simplest proceeds as

18,100BC final exam December 18, 2007 NAME: 1. (20 points) If S R, show that the set of isolated points of S is at most countable (use that the set of rational numbers is countable). Solution: Let I be the set of isolated points of S. If s I, let d

MODEL ANSWERS TO THE SECOND HOMEWORK 1. (a) is surjective. Indeed, let t R be a non-negative real number. Then t has a square root s R. In this case f (s) = s2 = t, by definition of the square root. is not injective. For example, (1) = (-1) = 1.

May 19, 2008 Math 220C Problem Set #6 Due: Wednesday, May 28 Grove, Chapter V: Exercise 1.2 on p. 172 Exercise 1.3 on p. 174 Exercise 1.4 on p. 176 Exercises 2.1 and 2.2 on p. 177 Exercise 3.1 on p. 182