NAME______________ Preactice Exam for MATH 3210 Directions: A main focus of this course is that each person develop the logical processes necessary to explain/prove their results in a clear, precise and logical fashion using relevant definitions. Your grade will be based on your explanations as well as the correct answers. Calculators are not allowed. Take your time and good luck. 1. (8 points) (a) Use truth tables to determine whether the following is a tautology: ( p (p q)) (q). (b) Is the converse of the above statement a tautology? Proof? 2. (7 points) (a) Let A be a n x n matrix with real numbers as entries. (a) Define the inverse of A. 2 1 4 3 2 5 (b) Let A = and let B = 0 1 1 7 5 3 3 2 2 . Use the definition in part (a) to determine 3 2 1 whether B is the inverse of the matrix A. 3. (10 points) (a) Let A = 2 3 -1 1 4 . Find A . (b) Find the matrix B such that 2 3 3 0 1 4 B= 1 2 1 NAME______________ 4. (8 points) Let A= {1, 3}, B = {1, 2, 5} and U = {1, 2, 3, 4, 5, 6, 7, 8, 9} (a) A X B (b) A B 5. (7 points) Let x be an integer. Use the direct method of proof to prove: If n is an odd integer then 5n + 6 is an odd integer . Explain all steps. 6. Let A = the set of courses required for your degree. Let the relation R be defined on A by course x is related to course y iff course x is a prerequisite of course y. Explain why this relation is a antisymmetric relation. Is this relation a transitive relation? Explain 7. (10 points) Show all computations for the following. Do not use a calculator. (a) How many bit strings are there of length six which are palindromes? Explain (b) How many bit strings are there of length 6 which contain exactly three 1's? Explain 7. (10 points) Suppose that a department contains 10 people, 4 men and 6 women. (i) How many ways are there to form a committee with 6 members, no restrictions? Explain. 2 NAME______________ (ii) How many ways are there to form a committee with 6 members if the committee must have more women than men? Explain. 8. Let A = { a, b, c, d } , and let r be the relation defined on A whose yes/no or Boolean matrix is a b c d a Mr = b c d 1 1 1 0 1 1 0 0 0 1 1 1 1 0 . 0 1 (a) (5 points) Draw the digraph of this relation. (b) (10 points) Assume the Boolean matrix above is Mr and that Mr represents the relation r where r represents the connecting flights that an airline has between 4 cities: a, b, c, and d. so there is a 1 in row x column y iff there is a connecting flight between (from) city x and (to)city y That is, the rows of the matrix represent the cities of the origins of the flight and the columns represent the destination cities. Let a stand for the airport in the city of Manchester, let b stand for the airport in Boston, c stand for the Chicago airport, d for the airport in the city of Denver. (i) Compute and interpret the Boolean products: Mr 2. Use the digraph in part (a) in your explanation. (Remember to use Boolean arithmetic) Mr 2 = Interpretation of Mr 2 (ii) Now call the given matrix A. Compute the product A2 using regular not Boolean arithmetic. Interpret what this means. Use the digraph in part (a) in your explanation. A2 = Interpretation of A 2. (iii) Explain path length with respect to this example 3 NAME______________ 2 9. ( 7 pts.) (a) Simplify 3 (i+2j) i=0 j=1 4 (b) 3 i=0 10. (8 points) Let A = {1, 2, 3} and B = {a, b, c, d}. (a)How many 1 - 1 functions are there from A to B? Explain. (b)How many functions are there from A to B? Explain. 10. (a)Assume that your job as a senior on campus is to assign students to dorm rooms. Assume you have nine students to assign to rooms and the rooms are a quad, a triple and a double. How many ways can you make the assignments? (b)Determine the coefficient of the term x2 y z3 in the expansion of (x + y + z)6 . (c) Determine the coefficient of the term x2 y8 in the expansion of (x + y)10 (d) Is the function f: R Z defined by f(n) = n 2 a one to one function? Is f an onto function? Explain. 4