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# 11Fe1s - K6 NAME Prof Girardi 09.27.11 Exam 1 Math 300 MARK...

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Unformatted text preview: K6 NAME: Prof. Girardi 09.27.11 Exam 1 Math 300 MARK BOX POINTS PROBLEM a; O (1) (2) (3) (4) (5) (6) (7) (8) (9) H b—-‘ Problem Inspiration Quiz 1 Exam 1 Fall 10 Number 3 Homework and Study Suggestion, § 1.1, 2 abdef Study Suggestion, § 1.1, He Study Suggestion. § 1.1, 9c Study Suggestion, § 1.2, 121) and Exam 1 Fall 10, # 6 Exam 1 Fall 10 # 7. Study Suggestion, § 1.2, 5c Exam 1 Fall 10 # 11. --- 100 INSTRUCTIONS: (1) The 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) This exam covers (from A Transition to Advanced Mathematics by Smith, Eggen, and St. Andre: 7th ed.): Sections 1.1, 1.2 . Numbers: 0 real numbers 2 R 0 natural numbers = N = {1, 2, 3,4, . . .} o integers = Z = {0, :l:1, :|:2, :l:3, i4, . . .} {qeRz p,qEZand q7é0} o rational numbers = Q Throughout this exam: (1) P, Q, and R are proposition forms. (2) P(a:), Q(.’L’), and R(a:) are open sentences with variable .77. 1. The Basics. Let P, Q, and 'R be ‘ﬁropositional variab‘I‘é’s‘. l ‘ m Theorem 1.1. l, in book * 1. P is equivalent to "" Q“ P) . (double negation law) * 2. P V Q is equivalent to (>1 V P . (commutative law) * 3. P /\ Q ' is equivalent to (9 1\ P . (commutative law) * 4. P V (Q V R) is equivalent to S P V (:1) V R . (associative law) * 5. P /\ (Q A R) is equivalent to ( P V\ (1L) A R . (associative law) * 6. [ P /\ (Q V R) ] is equivalent to l (P Y‘ H 2! (P V\ R1). (distributive law) * 7. [ P V( (Q /\ R) )] is equivalent to l f P V622 A ( P V R17 I (distributive law) ( ( *8. [ ~ (PAQ) ] is equivalent to “‘9 V 4 Q . *9. [~( (P)VQ] isequivalentto "‘ ‘7 V\"'g . Theorem 1.2.1 in book DeMorgan’s law) DeMorg‘an’s law) * 10. P : Q is equivalent to its contrapositive N Q :9 A“ P * 11. P => Q is not equivalent to its converse ()1 :5 P Theorem 1.2.2 in book Hints: Answers to (12) and (14) should NOT have =>. Answesr to (15) and (16) should have a 3. Also, your answers to (15) and (16) should be different from eachother. * 12. [ P => Q ] is equivalent to (A‘ P>V (>1 * 13. [ P 4:) Q ] is equivalent to [( PiDQ)“(C’L:3 P)j‘ (biconditional) * 14. [ ~ (P => Q) ] is equivalent to P N‘N Q) . (denial of implies) * 15. [ ~ (P /\ Q) ] is equivalent to P :5 "‘ (>1 . (denial of and) * 16. [ ~ (P /\ Q) ] is equivalent to :5 N P . (denial of and) * 17. [ P => (Q =>.R) ] is equivalent to K P A Q l :‘> R‘l‘l . * 18. [ P => (QAR) ] is equivalent to C! P962 if (P :5 RE * 19. [ (P VQ) => R] is equivalent to i( P QR)“ ( Q 13.—R)? 20. Circle true or false. . P50. 9:5 Q true @ P => Q implies P 4:) Q T T - - F GE? false P 4:) Q implies P => Q l; F u T “ @ Restore pruentliesos to the abbreviated propamitional form. er rim ; M J A , V J =£> 1 59> K [1, I: 7! R1 4: ( I; Q ) V (R /\ p ) :l j .3 ‘ O 1 1 ,1. .1 1) _ f __l l— ' J 1. ‘J‘ as---) A I L‘ ‘ 12h: 1r! \) I ‘L+ ~J:)_.,_____.__ ....m,......--—-——~—a .— ml. 9‘ 1 WW! / ”.1 O 3. Which of the following are propositions? Give the truth value of each proposition. Of course, justify your answer. 3a. What time is dinner? Noll o. '1 a ”my mm. . T141313oe 1C4+IUHI ¢J€Mf,lf\ 1L4— Amy'HL’mH/Jm Mli/ﬂ ”Kl A] l 1 ' . . . jrﬁﬁ‘. _{ "3 1-311, 3b. It 1s not the case that 5 + 7r is not a rational number. ~ (~19) -P rims is to» actuate measles law, bbgeb beeem us+P1LS and wewio're .5“ij +911»? 5+n‘ is m+fon0L\ whichm 3c. 2313+3y is a real number.l No u EI‘DQOSM myﬁ “013 \T) erwmwm / rib+ o pmeoxxwan e; bec. 01‘5”)“ 1.09 dew-l KNEW anaJ— X. “m“ (’4' Cﬂ C 3d. Either 3 + 71' is rational or 3 — 7r is rational. - 3 F1” 1g RJEﬁ}-ﬁ:-Ed :11 MW S’m. e 9;, k WE m 9““) Connected with on or‘ WW Wkly: is fudge, 3e. Either12 is rationaliand .7r is irrational, or‘27r is rational.1 -A:l.\—‘ 1.. yr 1 :1 (P A (231/ R {T /\ ‘T. \/ F~/j 4. A useful denial (NOT eginning with it is not the case that) of the statement a “Roses are red and violets are blue.” is : (1) W5 M‘C— Ml’ TCCl of viola-15 are r101- blag. m It roses are “Jim mom; on. «a blur, (35-74' “MOMS “3'6 HR") Him roses CU‘C- no+ N4 , Your answer in the above box should be a complete sentence. Rﬂges out NJ MA mabl's cu—c blue. ., Okla”- L____,____1 W 61 ~(P1a) (11 («42) v («(31) [Di/{0’31“} if“) P ;> («(1) M ‘5): mWem (3) a % (NP) “PM ‘ gLI‘FMC' C 5. Consider the statement [P A lel~PV ~Ql- 5a. Make a truth table for the propositional form in 5 . 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). contradiction neither Th0. East Wyn 1A 0.01 T (W63 Mr [+5 (2., bloloh 6. Show that (FAQ) ﬁR and (PA~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 xou. ’25 4:0 la '1" '1'} 333333 ans—1: adv—,— r-l I-JH VI ._1 "U '11 ’11 EB l%”.w.iizg 4,93 La; _ , 1 . A, ‘.- r) .- , 2 “-1 ,.\,‘ ff: l M t'._ ‘71., H'sv‘ " ' iv _: {’12- 7' i) ,4. ‘3“ -. “L'l’LCL (PP A A" P‘ ,1 “2/" ' ‘- 7. Fill in the blanks with P and Q as to provide translations of theﬁonditional P => Q. For example, If P , then Q o P implies Q . o 61 is necessary for j] . o P is sufﬁcient for a . o (2 if E . o P only if Q . 8. Is the conditional sentence - liliﬂhenls + 5 = 10.: F ‘1‘ false. Of course, explain your answer. Q Q fig F t ‘9. Circle T for True and F for False F There is a true conditional sentence for which the converse is true. F There is a true conditional sentence for which the converse is false. F There is a true conditional sentence for which the contrapositive is true. T ® There is a true conditional sentence for which the contrapositive is false. F There is a false conditional sentence for which the converse is true. There is a false conditional sentence for which the converse is false. T CF) T (9 There is a false conditional sentence for which the contrapositive is true. @ F There 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. 10. What has been your favorite problem/ example / idea/ class in the course thus far? Please write your CD Ms 1&1" 50W \Y‘fwl' .’ response on the back of this page. ...
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11Fe1s - K6 NAME Prof Girardi 09.27.11 Exam 1 Math 300 MARK...

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