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
Math 111C Homework #1
Due: Tuesday, April 17th (by 4pm to my office).
Instructions: Be sure to thoroughly justify all answers. I strongly recommend that you attempt all problems on your own before con
Math 111C Homework #3
Due: Thursday, May 3rd (by 4pm to my office).
Instructions: Be sure to thoroughly justify all answers. I strongly recommend that you attempt all problems on your own before consu
Math 111C Homework #4
Due: Tuesday, May 22nd (by 4pm to my office).
Instructions: Be sure to thoroughly justify all answers. I strongly recommend that you attempt all problems on your own before consu
Math 111C Homework #5
Due: Thursday, May 31st (by 4pm to my office).
Instructions: Be sure to thoroughly justify all answers. I strongly recommend that you attempt all problems on your own before cons
Math 111C Homework #2
Due: Thursday, April 26th (by 4pm to my office).
Instructions: Be sure to thoroughly justify all answers. I strongly recommend that you attempt all problems on your own before co
Mathl 1 1 C
Midterm
Time allotted: 80 minutes
Last Name:
1. Do not start until instructed.
2. Clearly explain your answer. Write neatly!
3.- If you run out of room, please use the back of the preced
MODEL ANSWERS TO THE THIRD HOMEWORK 1. 7. There are many ways to prove this. Perhaps the simplest proceeds as follows. Consider trying to construct a permutation f : S - S. Order the elements of S fro
MODEL ANSWERS TO THE FOURTH HOMEWORK 1. (a) No, this is not a group. The rule for multiplication is not associative. For example, Also there is no identity. 0 is a right identity, as a - 0 = a, but 0
MODEL ANSWERS TO THE FIFTH HOMEWORK 11. The solutions of the equation x2 = e are precisely the elements of G which are their own inverses. The function i : G - G is a bijection, since i is its own inv
MODEL ANSWERS TO THE FIRST HOMEWORK 1. Chapter 1, 1: 1. Suppose that a and b are elements of S. By rule (1) a b = a. But by rule (2), a b = b a. Applying rule (1) we get a b = b a = b. Thus a = a b =
MODEL ANSWERS TO THE SECOND HOMEWORK 1. Label the vertices of the square A, B, C, D, where we start at the top left hand corner and we go around the square clockwise. In particular A is opposite to C
MODEL ANSWERS TO THE THIRD HOMEWORK 1. (b) Circles centre the origin. (c) The real line union , where the number m R {} represents the slope. 2. Chapter 2, Section 4: 9. [0] = 0 + H = {[0], [4], [8]
MODEL ANSWERS TO THE FIFTH HOMEWORK 1. Chapter 3, Section 5: 1 (a) Yes. Given a and b Z, (ab) = [ab] = [a][b] = (a)(b). This map is clearly surjective but not injective. Indeed the kernel is easily s
MODEL ANSWERS TO THE SEVENTH HOMEWORK
1. For Chapter 2, Section 9: 1. Let : G1 G2 - G2 G1 be the homomorphism that sends (g1 , g2 ) to (g2 , g1 ). This is clearly a bijection. We check that it is
C m 3): Evan-cfw_MEL -mg hr: SW :E i
(3H2?) I: QM? Mi
m I . I. r f.
a 0 gm. k I C w w: ,
3
We; la. 8 :5 Hfh, Q (1 \y ("1 Fw_m\k 9mg :2:- EWWWW I3: (Ea-cfw_Way pm .:; Q 3 G F aw
H. 63.9%. (35291: (321