Solutions to Homework 9 - Math 2000 All solutions except 5.16 may be found in the book. We rst prove the following lemma: Lemma Let a Z. If 3 | a2 then 3 | a. Proof. (by contrapositive) Instead we shall prove a Z, if 3 a, then 3 a2 . Suppose 3 a. Then a =
Solutions to Homework 8 - Math 2000 All solutions except 4.22,4.24, and 4.36 may be found in the book. (# 4.22) (a) Prove that if r is a real number such that 0 < r < 1, then 1 4. r(1 r) (b) If the real number r in part (a) is an integer, is the implicati
Solutions to Homework 7 - Math 2000 The solutions to all exercises except 4.12 may be found in the back of the book. (# 4.12) Let a, b Z. Prove that if a2 + 2b2 0(mod 3), then either a and b are congruent to 0 modulo 3 or neither is congruent to 0 modulo
Solutions to Homework 6 - Math 2000 The solutions to 3.7,3.15,3.21 may be found in the book. Exercises 3.4,3.15 were done during the tutorial. (# 3.2) Let n N. Prove that if |n 1| + |n + 1| 1, then |n2 1| 4. Solution. Let S = Z and P (n), Q(n) be the open
Solutions to Homework 5 - Math 2000 The solutions to 2.39,2.43,2.45,2.49,2.53,2.55 may be found in the book. Exercises 2.46,2.48 (a),(c) were done during the tutorial. (# 2.40) Write negations of the following open sentences. (a) Either x = 0 or y = 0. (b
Solutions to Homework 4 - Math 2000 (# 2.18) Consider the open sentences P (n) : 5n + 3 is prime. and Q(n) : 7n + 1 is prime over the domain N. (a). State P (n) = Q(n) in words. (b). State P (2) = Q(2) in words. Is this statement true or false? (c). State
Solutions to Homework 3 - Math 2000 Notes: Check the back of the book for solutions to 2.1,2.5,2.9. 2.11 was done during tutorial. (# 2.2) Consider the sets A, B, C , and D dened below. Which of the following statements are true? Give an explanation for e
Solutions to Homework 2 - Math 2000 Notes: Questions 1.21,1.22 were covered in the tutorial. The solution to 1.23 may be found in the back of the book. (# 1.17) Let U = cfw_1, 3, . . . , 15 be the universal set, A = cfw_1, 5, 9, 13, and B = cfw_3, , 9, 15
Solutions to Homework 1 - Math 2000 Notes: I have removed 1.12(b) from the list. (# 1.4. (b),(d) Write each of the following sets by listing its elements within braces. (b). B = cfw_n Z | n2 < 5 (d). D = cfw_x Z | x2 x = 0 Solution. (b). Note that n2 < 5