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Unformatted text preview: Sample Tests 405 Sample Tests This section of the Instructor’s Guide contains sample tests for an introductory course in discrete mathematics. Two tests are included for each chapter of the text. The problems on these tests were used on examinations given in discrete mathematics courses at various schools, or are similar to such questions. The first test contains straightforward problems and is easier than the second test. Some of the problems from these second tests are moderately difficult. I have also included two sample final examinations. The second of these is the more challenging examination. You may want to use these tests as a source of questions for your own examinations, rather than using them exactly as they are. If you do so, select questions primarily from the first of the two examinations for straightforward questions, and from the second for more challenging questions. Also, for a much richer set of questions, consult the extensive test bank also included in this Guide . These sample tests are an attempt to test students efficiently. Wherever appropriate, problems with numerical or short answers are given. However, there are many places in the course where it is not possible to assess students adequately without requiring longer answers. You will find that there are several problems where I have asked students to prove or disprove a statement. I find that questions of this sort test whether students can think mathematically and write correct mathematical arguments. On my examinations I give explicit directions to students to provide complete answers, including reasons for the steps of proofs; I advise you to do the same. Each sample test has been printed on its individual page. Solutions are provided imme diately following the test. 406 Sample Tests Chapter 1—Test 1 1. What is the truth value of ( p ∨ q ) → ( p ∧ q ) when both p and q are false? 2. What are the converse and contrapositive of the statement “If it is sunny, then I will go swimming”? 3. Show that ¬ ( p ∨ ¬ q ) and q ∧ ¬ p are equivalent (a) using a truth table. (b) using logical equivalences. 4. Suppose that Q ( x ) is the statement “ x + 1 = 2 x .” What are the truth values of ∀ xQ ( x ) and ∃ xQ ( x )? 5. Prove each of the following statements. (a) The sum of two even integers is always even. (b) The sum of an even integer and an odd integer is always odd. 6. Prove that there are no solutions in positive integers to the equation x 4 + y 4 = 100. Sample Tests 407 Chapter 1—Test 1 Solutions 1. When p and q are both false, so are ( p ∨ q ) and ( p ∧ q ). Hence ( p ∨ q ) → ( p ∧ q ) is true. 2. The converse of the statement is “If I go swimming, then it is sunny.” The contrapositive of the statement is “If I do not go swimming, then it is not sunny.” 3. (a) We have the following truth table....
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 Spring '09
 Egiceoclu

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