3. If today is Tuesday, what day of the week will it be 900 days from now? 3. (8 points) Consider the Venn diagram shown below. Shade the region corresponding to the set (BUC) - A. (a) Saturday (c) Thursday (e) Sunday (b) Wednesday (d) Friday 4. Calculate the following sum - n+1 , 49 (a) 99 100 (C) 200 49 99 (b) 100 (d) 200 5. The number 4. (12 points) Let A = {n E Z | n = 5r for some integer r} and B = {m E Z | m = 0.737373 ... 20s for some integer s}. expressed as the quotient of two integers is (a) Prove that A & B 737373 73 1000000 (c) 99 1.3 (d) 73 b) 9.9 100 Free Response 1. (8 points) Consider the recursively defined sequence: S1 = 1, S2 = 1, Sk = k . Sk-1 - Sk-2, for all integers n 2 3. (b) Prove that B C A Find S3 and S4. S3 = SA = 2. (12 points) Let A = {a, c, g, h, i} and B = {a, d, i} with universal set U = {a, b, c, d, e, f, g, h, i}. Find (a) AUB = (b) AC = (c) An B =5. (10 points) Show by contraposition the following statement. Use the definitions of the 6. (10 points) We want to prove the following theorem by mathematical induction: terms. Do not use any other facts previously proved in class or in the text or in the exercises. "P(n) : For all n 2 1, we have that 1 + 5 + 9 + 13 + . . . + (4n - 3) = n(2n - 1)" Given n an integer, if (n + 3)2 is even then n is odd. Suppose we already proved the basis step. By inductive hypothesis we have that P(k) is true. That is 1 + 5 +9 + 13 + . . . + (4k - 3) = k(2k - 1). Your task is to prove that P(k + 1) is true. That is: 1 +5+9+ 13 + . .. + (4k - 3) + (4k + 1) = (k + 1)(2k + 1).Page 1 of 5 Discrete Mathematics MATH1071, Test #2 Name: General Directions: All work must be shown, except true or false, and multiple choice questions. Partial credit will be given as warranted. Cell phones must be turned off and put away. True or False. Write the word true or the word false. Each question is worth 3 points. 1. Let p and q be two integers. Then 6 | 3p(8 - 4q). 2. For each integer n, n 2 2, let P(n) be the formula [i(it1) = n(n-1)(n+1) 3 Then, P(2) is true. 3. Consider the sets A = {2,31 mod 7, 4}, and B = {14 mod 6, 3}. Then BnA = {2}. 4. Let A and B be two sets, with A C B. Then, An B = A. 5. V4 + 3 is irrational. Multiple Choice: Each question is worth 5 points. Circle the right answer 1. Compute the following: 1000! 998! (a) 1000 (b) 1000/999 (c) 999000 d) Undefined 2. Suppose a is an integer. If a mod 7 = 4, then 5a mod 7 = (a) 2 (c) 3 (e) 6 (b) 4 (d) 5