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# Exam1tornsol - Math Csci 2610 Fall 2004 Instructor Zubeyir...

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Unformatted text preview: Math / Csci 2610 Fall 2004 Instructor: Zubeyir Cinkir Exam 1 Date: Sep. 16, 2003 Duration: 50 min Direction: For full credit, write down your arguments as clearly as possible and in an organized way, (so that also it can be used to give a partial credit). Name: Question#l: (20pts) Answer the followings: 1. (10 pts) Show that (p EB q) ——) (p V q)is a tautology by using truth table. 2. (10 pts) By the logical identities, show that the propositional compound (-'q /\p) —) (p V q) is a tautology. Harsh“. Question#2: (10pts) If It is rainy, then I will stay home. 1. (5 pts) Write the converse of this statement. 2. (5 pts) Write the contrapositive of this statement. - * "trot-\ﬁtj' me MN.“ ‘5‘. ‘5 no 2- :12 1 W ““3“” ' Question#3: (10pts) 1. (5 pts) Let A(x) be the statement x lives in Atlanta Let F(x,y) be the statement x and y are friends Assume that the universe of discourse consists of all students at UGA. Translate the following into English VX3Y(A(X) V (MY) /\ F(x,y)”- 2. (5 pts) Write the negation of the statement in 1. Then, translate it into English. i- UGR I H5211: is 01 siuienli‘xjmjt U63 or )4 has a grim}; x) who is 01 Question#4: (15pts) Let (t) n is an odd integer. (it) 5n + 4 is an odd. 1. (7 pts) prove that (i) —) (ii). 2. (7 pts) prove that (it) —) (t). (1 pts) After these two, what is the conculsion about (i) and (ii) k is 04:3,. Than} {1: 1k+\$ g...— arc-M9. M55130!“ 1 1+ : \O‘F'ta‘ :l(§k+f+)+2\ 3(1k'1‘q1rl‘t : (DY—45+ okksn “m 9 5thl+ : . wt; gn'th ‘5 a “ﬁzzy-(swc V“ N 3 ) ' J whoa. gk'tl't ‘8 “A .) “rm, was 03'a [“ 5" MM) }f\5,-c_3e,r “a“ on“; gm“ C Se; ?W%.\$ b? M. n ﬁxerC-ise' Q__ 1 \gan. “A“ 5 a Question#5: (18pts) Let A: {(1, b, c, d, e} B: {1, a, b, 2, c} C: {3, c, a, 1 , f} 1. (3pts)WritethesetAnB. = ROMEC} 2. (3pts)WriteAUB. e §mtb.c.d.6. «(2.77 3. (3pts)WriteA-(BﬂC) = A-X’hﬂl'ﬂl ": S’E'A‘e'f C g) ((5) (9.5:); 4. (3pts)Write(AﬂC)><(C-B) =§a,c'1%‘°\”>;§1f =3] (“'3)’ a" ' i I 5 1 5. (3 pts) What is the cardinality of P(A),power set of A? [A kg "" \?(N\:2 :3 6. (3 pts) What is the cardinality ofA xC? * m=i 5,, web \Cﬁ : ‘3 Question-#6: (10pts) Let A and B be given two sets. Show that A U (B-A): A U B . 9mg M \oamt’f Wu“ T \e, ave W B as ﬁlﬂ 8.x Vex-Laﬁ- nobﬂgk" ' Question#7: (17pts) Let sz —> B V x E AV y e A ((f(x) = f(y)) —) x = y) is true, we say that the function is injective. 1. (3 pts) By using quantiﬁers, as above, state the deﬁnition of sujectivity for this func- tion f:A —) B Now suppose A=B=set of integers and f is given by f(x)=5x-3 2. (7 pts) Is f injective? Give a complete argument supporting your answer. 3. (7 pts) Is f surjective? Give a complete argument supporting your answer. \qt bee; 3.“er (““350 gm=§ii~53 9"" ”M3 i xgx—b. \WI *5 at. ”the“; 9455—3 g, :5 “Scat-Nb Question#8: (10pts) (Bonus) For any real numbers x, show that the following equality holds. L2xJ=LxJ+Lx+%J that 12M 3&4“;- ‘Thif wan; an excitwuﬂe 5a\uWe.L M ‘9‘?— Question#9: (5pts) (Bonus) Write the negation of the following statement VX(P(X)) —> Vx3y(P(X) Aanl) Fxgwcﬁe \ ...
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Exam1tornsol - Math Csci 2610 Fall 2004 Instructor Zubeyir...

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