Homework Answers

• 3.3 (m*n)*t = m*t = m; m*(n*t)= m*n =m so associative. Not comm.
• 3.5 (m*n)*t = (m*n)^2 + t^2 = (m^2 + n^2)^2 +t^2 m*(n*t)= m^2 + (n^2 + t^2)^2. Not assoc. Clearly comm.
• 3.13 a and c are true. b may not be.
• 3.19 even positive integers.
• 3.22 only one comm operations on a one element set; eight = 2^3 cmm operations on a 2 element set. Have two choices for x*x, x*y and y*y. For a three element set have 3 choices for x*x, y*y, z*z, x*y, x*z, y*z. S0 3^6 choices.

SAMPLE EXAM II

• Problem 1. Define a relation R on the real numbers as follows: say aRb if a less than or equal to b. Which of the properties of an equivalence relation are satisfied and which if any are not.
• Problem 2. Let m = 440 and n = 126. Give the prime factorization for each. What is the gcd and lcm of 440 and 126.
• Problem 3. If (a, m) = 1 and (b, m) =1 prove (ab, m) =1.
• Problem 4. If phi is the Euler phi function, prove phi(3n) is either 2 times phi(n) or 3 times phi(n).
• Problem 5. Let G be a cyclic group of order 36 with genrator g, so G = . Find all generators of the cyclic subgrup of order 6.

  FINAL

  • My understanding is that our final will be given Saturday, Dec. 13, 10:30 a.m.-12:30 p.m.