MATH 233 Extra Problems for the Final Exam A. Relations 1. Suppl. Exercises for Ch.7 (p.379): 4, 7, 8, 16a, 18ab. 2. Let A = {a, b, c, d, e, f, g}. Find the number of relations R on A such that (a) R is reflexive, symmetric, (c, f ) R and (g, d) 6 R; (b) R is reflexive, antisymmetric, and |R| = 9 (i.e. R has exactly 9 elements). 3. Let S = {d N | d|60 and d > 3} (divisors of 60 which are greater than 3). Define a partial order R on S by xRy if x|y. Draw the Hasse diagram of R. Find minimal, maximal, least and greatest elements in S. 4. Let A = {a, b, c, d, e, f, g, h} and B = {a, b, c, d}. Let P(A) be the set of all subsets of A and define the relation R on P(A) by (X, Y ) R if X B = Y B. (a) Verify that R is an equivalence relation. (b) Find the number of elements in the equivalence class [{a, c, e}]. B. Rings 1. Sect. 14.1 6, 7, 9bcd, 14, Sect. 14.2 3a, 4, 5, 11, 13, Sect. 14.3 5, 32, Sect. 14.4 3, 9, 10. Suppl. Exercises for Ch. 14 (p. 708) 1, 2, 3, 8. 2. Find all values of m for which the set Q with the operations and defined by a b = a + b + m and a b = a + b + 7ab is a ring. Determine if this ring has a unity and zero divisors. 3. Is it true that if a b (mod n) then 7a 7b (mod 7n)? Justify your answer. 4. Find the multiplicative inverses of [3] in Z8 , Z10 and Z15 (If it does not exist, explain why.) 5. List all subrings of Z15 . Which of those subrings are ideals? 6. Determine whether the following rings are isomorphic and justify your answer. (a) Z4 Z6 and Z24 ; (b) Z5 Z6 and Z3 Z10 . 7. Find all values of a and b, such that a > 1, b > 1, for which the ring Z100 is isomorphic to Za Zb . 8. HW problems for 17.1-2 C. Groups 1. Sect. 16.1 1, 3, 4, 5, 8, 9, 10, 11, 18; Sect. 16.2 3, 7, 9, 10, 11, 15; Sect. 16.3 1, 4, 5, 6; Suppl. Exercises for Ch. 16 (p. 797) 1a, 3. 2. Check whether the following are groups: (a) The set Z with the operation x y = x + y + 5. (b) The set Q with the operation x y = x + y + 2xy. (c) The set [1, ) = {x R | x 1} with the multiplication operation. 3. Let H be the set of all elements g in the permutation group S3 of order 1 or 2. Determine if H is a subgroup. 4. List all the subgroups in the group of symmetries of an equilateral triangle. Is this group cyclic? Explain. 5. Are the groups G = Z12 and H = Z2 Z6 isomorphic? Justify your answer. 6. Show that if G is an abelian group, then the map f : G G, f (g) = g 2 is a homomorphism. 7. If G is a non-cyclic group of order 30, what is the largest possible order elements of G may have? \u0012 \u0013 1 2 3 4 5 6 7 8. (a) Find the order of the permutation in S7 . 5 4 7 2 6 3 1 (b) Find the largest order of an element g in S7 . 9. (a) Find 4375 (mod 37); (c) Find the last digit of 7777 ; 10. Review the last hw. (b) Find the remainder of 225 + 367 after dividing by 23; (d) Find the last two digits of 97122