mt1key - .- Name; K E’ g MATH 108 ‘ Section 1 First...

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Unformatted text preview: .- Name; K E’ g MATH 108 ‘ Section 1 First Midterm October 22, 2010 Answer all questions on these pages. If you need more space, write on the back of the previous page and indicate that you have done so. Since this course emphasizes the basics and rigor, your - justification of an answer is more important than the answer - so always give a clear and complete justification unless you are instructed not to do so. 1. In each part below, give the definition: (15) a. The conjunction of propositions P and Q. t, i \ kl Two =51“th HP one! Q '. we wr{ PAz-SL, ma m T”T‘°"‘+‘°‘a~ is +er who“ ‘3 3‘4“ P (SeikoL 61 can?» CIA/L «Cu \S‘; o‘H~z-VW'\5V¢ - b. r The—intersection over a family of sets 71. AC)? A :I X )(VAé‘eNXeME 0- Ammdexed family of sets. k6 7' éAq i “59% i9 TCQVWCCQ JtSSc-im‘i' V d/FQD ’v‘fflxaflg m“ AomAlgch 2. Indicate if each of the following statements is true or false. If a statement is true, prove it directly (do not cite any theorems). If it is false, give a counter example: (15) a. With N as the universe, (3n)(n2 + n + 41 is prime). Lei MI, I‘M-widthcrza raw b. WithNas the universe, (Vn)(n2+ 11 +41 is prime). F; Lei “at, Lti‘wwri: LHUH +l+|>>+ws No~+ 7am. c. ForsetsA,B,andC,(AnC=BnC)=>(A=B). V: w.» Am». c: £23 me» An balsam B: €253 EVA R43. 3. a. Negate the following expression: (3x)(Vy)(P(x,y) A R(x) => (Elz)(Q(z) v S(z))) <12) ( m (3 p (POW) /\ mm (V2) (~a (am/9)) (Problem 3 continues on next page) b. In the universe of Q, give proofs of the following theorems: _ i- (VY) (3X) (X - y > 0) ‘Fo r‘ an curioi in V7; )1 ~[(ax)(Vy)(x-y>0)1 <9 (vi) (3,) (X’s/so) gc/“Cur ck CJ<VQM X Ive—F \/ ': TM“ >97: X*C>L—H)'; vL’X*l‘=~'\ so, 4. a. State the tautology that justifies (14) ,,4_4fi°rtwe-partiffpr°°fiwifl?€5 Gil (“:9 l ( Fr» Q) /\ (Q9 W] Proof of a conditional statement by contradiction: (Pt; Q) < :2) [ (l’ch'Q) -:>({z A442?) b. Prove that if x2 is not divisible by 4 then x is odd. U 2 Z Use vatm fog-1410 n . C17’. x avert 1-.) x7— Altvzsilale 1’1 “t , V10”: x22}; torssmlz gnaw :“rk‘ , icucsclole» L1” . *; , 5. a. Let A = {a, {a}, E, {b}}. How many elements are in the power set ‘P(A): ; ‘36 In each case below, circle any of the three symbols in the brackets that applies: (2°) i. z @gQA. ii. a @c,g] A. iii. {z} [e,©@ A. iv. 6 ’.P(A). (Problem 5 continues on next page) b. HA and B are sets,prove that AQB =9 CP(A) Q ’P(B). \fx («C X€P(M thwx XsA. gm x9; an» Ad; go xeflB) g0 WOMEN/05> Kéflgw v» we 2— Hg) . 6. Define the sequence {an} as follows: al‘ = 1, a2 = l, and and: an+1 + a,I + an+1 a,. for n z 1. B10me gravy-*1], Lwherefi f,I is then?" Fibonacci number. (12) «FCL’DGKRCL‘ " ‘/‘|/ 1/;lgl " ~ - :41“! +£Dx—1\ i “.:l=z”\ U9¢\7C‘£, “1:171." . ,n ’ Aésum P(l)/w./PL \kl/f\ ah“: ah+am+qnaw A t,“ :2 ". . '2 x— m (fun 0% a) - PM. L37 P c 1 . 7. Prove that WOP => PMI. ’ (12) arrow SCAN MA awes‘ 3;)neS '27 h—H as Vhé/Ui Le+ (B'=§\IN, A94uw. g4 Tken B ow (\[e.q;+’ {(§MV\+ L W01), 5517 M» Bcul' M'i‘ €iAC2-‘©1‘;/,£—SI r Le+ h=m4 I TM keS 5'14“, k<Mmuiwxfs “hi/“(05“ 21*.043 der/ L317: k egg) RHQS an (m-\\ "'193 or mega ‘Tkic 79 EL MKMJIJ—Lm +5 “A673 80 B: 0% “wk 91w, ...
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mt1key - .- Name; K E’ g MATH 108 ‘ Section 1 First...

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