NAME: Prof. Girardi 09.27.11 Exam 1 Math 300 problem MARK BOX points 1 40 2 5 3 10 4 5 5 10 6 10 7 5 8 5 9 8 10 2 total 100 Problem Inspiration (1) Quiz 1 (2) Exam 1 all 10 Number 3 (3) Homework and Study Suggestion, 1.1, 2 abdef (4) Study Suggestion, 1.1, 11e (5) Study Suggestion, 1.1, 9c (6) Study Suggestion, 1.2, 12b and Exam 1 all 10, # 6 (7) Exam 1 all 10 # 7. (8) Study Suggestion, 1.2, 5c (9) Exam 1 all 10 # 11. INSRUCIONS: (1) he mark box indicates the problems along with their points. Check that your copy of the exam has all of the problems. (2) When applicable put your answer on/in the line/box provided. Show your work UNDER the provided line/box. If no such line/box is provided, then box your answer. Explain your answer when needed (if in doubt if needed, ask Prof. Girardi). (3) During this exam, do not leave your seat without permission. If you have a question, raise your hand. When you finish: turn your exam over, put your pencil down, raise your hand. (4) You may not use a calculator, books, personal notes. (5) his exam covers (from A ransition to Advanced Mathematics by Smith, Eggen, and St. Andre: 7 th ed.): Sections 1.1, 1.2. Numbers: real numbers = R natural numbers = N = {1, 2, 3, 4,...} integers = Z = {0, ±1, ±2, { ±3, ±4,...} } rational numbers = Q = p q R: p, q Z and q 0 hroughout this exam: (1) P, Q, and R are proposition forms. (2) P (x), Q(x), and R(x) are open sentences with variable x. 1
1. he Basics. Let P, Q, and R be propositional variables. heorem 1.1.1 in book 1. P is equivalent to. (double negation law) 2. P Q is equivalent to. (commutative law) 3. P Q is equivalent to. (commutative law) 4. P (Q R) is equivalent to. (associative law) 5. P (Q R) is equivalent to. (associative law) 6. [ P (Q R) ] is equivalent to. (distributive law) 7. [ P (Q R) ] is equivalent to. (distributive law) 8. [ (P Q) ] is equivalent to. (DeMorgan s law) 9. [ (P Q) ] is equivalent to. (DeMorgan s law) heorem 1.2.1 in book 10. P Q is equivalent to its contrapositive. 11. P Q is not equivalent to its converse. heorem 1.2.2 in book Hints: Answers to (12) and (14) should NO have. Answesr to (15) and (16) should have a. Also, your answers to (15) and (16) should be different from eachother. 12. [ P Q ] is equivalent to 13. [ P Q ] is equivalent to. (biconditional) 14. [ (P Q) ] is equivalent to. (denial of implies) 15. [ (P Q) ] is equivalent to. (denial of and) 16. [ (P Q) ] is equivalent to. (denial of and) 17. [ P (Q R) ] is equivalent to. 18. [ P (Q R) ] is equivalent to. 19. [ (P Q) R ] is equivalent to. 20. Circle true or false. true false P Q implies P Q true false P Q implies P Q 2
2. Restore parentheses to the abbreviated propositional form. P R = Q R P 3. Which of the following are propositions? Give the truth value of each proposition. Of course, justify your answer. 3a. What time is dinner? 3b. It is not the case that 5 + π is not a rational number. 3c. 2x + 3y is a real number. 3d. Either 3 + π is rational or 3 π is rational. 3e. Either 2 is rational and π is irrational, or 2π is rational. 4. A useful denial (NO beginning with it is not the case that) of the statement Roses are red and violets are blue. is : Your answer in the above box should be a complete sentence. 3
5. Consider the statement 5a. Make a truth table for the propositional form in (5). [P Q] [ P Q]. (5) P Q 5b. Is the propositonal form in (5) a tautology, a contradiction, or neither? Circle one below and then explain your answer in a sentence (or two). tautology contradiction neither 6. Show that (P Q) = R and (P R) = Q are equivalent by making a truth table and explaining (in words) how the truth table tells you what you want it to tell you. P Q R 4
7. ill in the blanks with P and Q as to provide translations of the biconditional P Q. or example, If P, then Q. implies. is necessary for. is sufficient for. if. only if. 8. Is the conditional sentence true or false. Of course, explain your answer. If 7 + 6 = 14 then 5 + 5 = 10. 9. Circle for rue and for alse here is a true conditional sentence for which the converse is true. here is a true conditional sentence for which the converse is false. here is a true conditional sentence for which the contrapositive is true. here is a true conditional sentence for which the contrapositive is false. here is a false conditional sentence for which the converse is true. here is a false conditional sentence for which the converse is false. here is a false conditional sentence for which the contrapositive is true. here is a false conditional sentence for which the contrapositive is false. You might want to (but do not have to) use the below chart to help you figure out the answers to number 9. P Q 10. What has been your favorite problem/example/idea/class in the course thus far? Please write your response on the back of this page. 5