HOMEWORK SOLUTIONS 02/05/03
6.1(b,c)
Find ve integers solving the congruence relations, at least one of each negative:
First, n 1 (mod 10). Its easy to see that n = 9, 11, 21, 31, and 29 all work.
Then, n 8 (mod 12); n = 4, 20, 32, 44, and 56 all work.
6.
Name:
Math 112
First Midterm Exam
Instructions: This exam has a total of 100 points. You have 50 minutes. Please show every step, no matter how excruciating, and give the reason why it works - axiom or theorem or whatever. Please use bluebooks, and start
SECOND MIDTERM REVIEW
1. Overall Info You will have 50 minutes for the exam. There will once again be a combination of `computational' problems and more theoretical proofs. The format will be extremely similar to the first test, though probably with a lit
FIRST MIDTERM REVIEW
1. Overall Info You will have 50 minutes for the exam. There will be a combination of `computational' problems and more theoretical proofs. Most of the proofs will be taken directly from the text or the homework; some of the others mi
HOMEWORK SOLUTIONS 12/05/03
6.8
Make a multiplication table for U (9). Is this a cyclic group? If so, nd all generators.
First of all, the set of U (9) is only those 1 a < 9 such that (a, 9) = 1. This set
is cfw_1, 2, 4, 5, 7, 8. Here is the table:
1
2
4
HOMEWORK SOLUTIONS 11/04/03
2.1 Using the axioms or results, prove that -(a + b) = (-a) + (-b) for all a and b. I will show two ways to do this. First, recall that -(a + b) is the additive inverse of a + b, so we only have to show that (-a) + (-b) is also
FINAL REVIEW
1. Overall Info You will have two hours for the exam. There will once again be a combination of `computational' problems, more theoretical proofs, and some short (or just) answer questions. It should lie between the two midterms in difficulty
HOMEWORK SOLUTIONS 09/04/03
1.11
We have two functions, f : A B and g : B A, as described in the text. Since
the range of g is the domain of f , we can create f g, and its range is the same as
the range of f , since that is applied last - that is, B. Simi