06Sfs - f} NAME: 2 ’3 -' Prof. Girardi 04.27.06 Final...

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Unformatted text preview: f} NAME: 2 ’3 -' Prof. Girardi 04.27.06 Final Exam Math 300 |—_ MARK BOX PROBLEM POINTS Problem Inspiration 1 I— 11 _l— Ease handout 2 12 class handout 3 5 I § 1.2 # 7e 4 - 1st chokgl 9 4a: hand—in HW § 1.5 # 3g 41): look-at HW § 1.6 # 7k 4c: iii—class HW 1.7 # 3d 4d: Theorem 2.6 n 4 - 2nd choice 9 —l |_4 - 3rd choicd— 9 | | F TOTAL 55 _l_ | 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/hex. If no such line/box is provided, then box your answer. (3) For problems that request a proof, write a neat FORMAL proof and use only logic and the definitions of the concepts involved (i.e., do not quote a problem from the book). You may include your skeleton, if you so wish. (4) During this exam, do not leave your seat. Ifyou have a question, raise your hand. When you finish: turn your exam ever, put. your pencil down, and raise your hand. (5) You may not use a calculator, books, personal notes. (6) This exam covers (from A Mnsition to Advanced Mathematics by Smith, Eggen, and St. Andre: nth ed.): Sections 1.1., 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 2.1, 2.2 . Numbers: a real numbers = R 0 positive real numbers 7— ll?“ : {:12 E R: :1: > 0} 0 natural numbers = N = {1, 2, 3, 4, . . o integers = Z = {0, :|:1, 2, :|:3, 4, . . .} o rational numbers = Q = {E E "t: p, q E Z and q 7é U} Throughout this exam: (1) P, Q, and R be propositions. (2) P(:r:1, . . . , 11:7,), (201:1, . . . , 311,), and R(:rl, . . . ,mn) are open sentences with variables 51:1, . . . ,atn. 1. Make true statements by filling in the blanks with the appropriate number between 1 and 11 from below. Use each number once, and only once. m (P => mm? => 1:) pm? = Q)/\(Q =¢ p) ,6) ~ Q => ~ P MUD/x m Q) => R M (M ~ Q) in ( ~ P) A (~ 62) {WP => Home =2 R) £8) P =~Q l0? (Vx)[~ Pm] ()0) (39:)[~ FLT-)1 (11) (EmJlPWJ/WWNPW) => :6:le P => Q is equivalent to 3 P <=> Q is equivalent to 2 m (P =- Q) is equivalent to w (P A Q) is equivalent to m ( P V Q) is equivalent to G) ____ P 2 (Q A R) is equivalent to P => {Q V R) is equivalent to (P V Q) i R is equivalent to " m [(Vm)P(m)] is equivalent to I} ~ [(3:r)P(.r)] is equivalent to 9 (3!:c)P(:n) is equivalent to l l 2' Circle T if and only if the statement is TRUE. Circle F if and only if the statement if FALSE. 1. @ F (Fm)(vy)F(x,y) :> (chvmumy) 2» F (FFMFMM) => (ayuamxw) F {Var)P(:F) (Elm)P(m) ’4 T if? (3F)P(rc) => (mph) <3 T (VF)[P(F)VQ(F)] =» [(ijmmwwmxm 3 F [(vmwmwwxmtxn => wasnptxwmxn L1 F (WM) :> em] => mm.) => (VFWCFJI “1 T [(vx)F(ar) => (qume = MW) => mam F F‘ (Mammy => (EFH‘F’mey) {:1 F (ax)(v:u)F(x,y) => (Fyxaxjpw F (Vm)[P($)/\Q(m)] => [(VF)P(F)MVF)Q(F)] '3 F [wanmeme => (VF)[P(F)/\Q(F)] 3. Show that N (P =- Q) is equivalent to (PA N Q) by making a truth table and then writting a sentence or so about how you are remng your truth table to come up with your answer. 4. Do THREE problems from problems: 4a, 4b, 4c, and 4d. Fill-in the blanks on this sheet of paper but include your proof parts on a separate sheet of paper (with your name on each sheet please). Circle THREE: I am doing problems 4a , 4b , 4e , 4d 4a. Let :1: E Z. If 8 does not divide x2 — 1, then m is evefl a. Let a, b, e E Z. By definition: .- 5‘:- l' v--.-' I . '\ a divides b, denoted by elb, ifand only if [ 3 -' A l l i ‘ " J J ‘ “I: j I) n f- l, w; .l c is even if and only if l 3 r‘C 51.! j l a. ’ “-' “‘- x cisoddifandonlyif (Cr 9? o. Symbolieally write, using quantifiers, statement 4a. ., ’ ‘Ir’l f I: _ ' ‘ i’ , '1 \' a}r >5 .r' *-.- ' .u l w -4“ m . 1-} E c “I Ii =_\ .2 ' _ ..— , ,-. Ml seesaw o._ Prove statement 43.. o. By definition, an integer M is odd if and only if ' -- ' :- 31'“? i 0. Symbolically write, using quantifiers, statement 41). I I :' E M E“ l ‘r :7 . :' fr" 7"; 5%.: (f 0" O I} \ o. Prove statement 4b. 0: 40. For each rational number :1: and each irrational number 3;, there is a unique irrational number 2 such that y + z = r. 0. Let a and b be real numbers. By definition: ” . 4 ...' I ._ Ir | a is rational if and only if ( 3 l3 Ii 9: .' l ‘ b is irrational if and only if o. Synibolieally write, using quantifiers, statement 4e. .1“! ' l a" i ‘11 v. .2 ' _ l 31. '7: a lit» 11L) .. if 3""— ' _—L L_. o. Prove statement 4e. You must carefully prove, using only the definitions above. (and not using another homework problem), that your 2 is indeed irrational. 4d. Let A, B, C be subsets of the universe U. Show that A U (B fl 0) = (A U B) n (A U C) o. By definition: :‘i'i :1: e A o B if and only if .-;-r rt x mEBflCifandonlyif a. Let P, Q, and R be propositions. The disbributive rule says that (Mm) v ("H R: [P A (Q V R)] is logically equivalent to [P v (Q /\ R)] is logically equivalent to (I? V Q) A ( v R) o. Prove statement 4d. You can use the logical equivalents for propositions that you just wrote above. I“. Ff 1.? W wwfl— ASSlee. 7‘ Hy). 1'! 0C 5 Lei. LA ’ (“3132113- #1: l..._ "(I ' \‘L ‘1 art“.- 1 , 4 ‘ _ '- -: ‘1 fl "" f _.. L2: ' r :1 ‘l ‘ \ll - 1: Li'r J I ['1 L-J jg ‘1' I . v F ' -' f; ~=. IJ‘ M I'r‘in y:- r I —; 1-} k-"'§t+lj :‘ “5 3-3 7: :1 . _ . | L m aj .4111 ‘1 d {3 M mil—I} if fi're"1,.- F 7 F“ # (tax {EMo-_ 1| -1 — ind: I r . ('1 J, —11 r :F¢fla\\ Mar F) é: \ m ,‘Ji" .- J‘ 1 I It r :5, L2“ ‘ (I '.-.-=. (J I? Z = 'JC"‘-‘. Fl“ L" 5 emf: >1. 5 (Q . t MFA/l‘+gvr(- Lg-é £7— (Xfflg E 4 _—_ x w—L LL'Uv‘cCTE jab _ ' ‘ u mu.— -.1[=.a.—Jt Z 5 {7Q 36 (iiiflhuf. 2L 5 63. 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06Sfs - f} NAME: 2 ’3 -' Prof. Girardi 04.27.06 Final...

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