MATH 211 Midterm, Fall 2005

AMERICAN UNIVERSITY OF BEIRUT Faculty of Arts and Sciences Mathematics Department MATH 211 MID-TERM EXAMINATION Fall 2005-2006 Closed Book, 1IH15MIN WRITE YOUR ANSWERS ON THE QUESTION SHEET | STUDENT NAME iD NUMBER {Problem | Out of 1 10 | 2 10 10 | 10 1. (10 points) {a} (5 points) Prove or disprove : (p> (gar) = (Usp g)A(pvr)). (b) (5 points) Consider the following sets: R = {{1}. {1.3}. {1.3.5}}. 5 = {1.2.3}. T = {1.3}. What are : i. (2 points) RU 2° (where 2° is the power set of S)? ii. (3 points) (S x T)U(T x S)? 2 2. (10 points) Solve the following : (a) (5 points) Vn € N(6|n? if and only if 6|7). (b) (5 points) Prove that V6 is irrational. 3. (10 points) Given the universal set of the days of the week and the predicates : w(d) 2 “Tl work on day d”, s(¢) :“it is sunny on day d”. p(d) :* partly sunny on day d”. write the following hypotheses in terms of logical connectives on r. f.s.d.t. Hy: “If I work on a day then it is either sunny or partly sunny”. Hy: “YT worked last Sunday or 1] worked last Friday”. Hy: “It was not sunny on Tuesday”. Hy: “It was not partly sunny on Friday’. Draw your conclusion. Indicate the rules of inferences you are using, L. {10 points) (a) (5 points) State the necessary condition for an integer to be composite (not prime). (b) (3 points) Check whether 143 and 1001 are prime integers ? (e} (2 points) Find the primes decomposition of 247. 5. (10 points) Let f(v) = 2/2] - FEl.n eZ. (a) (5 points) Using the Euclidean division theorem n = 3 x # +r. find out the range (image) of the function f. (b) (2 points) Find f71({O}). the inverse image set of {0}. (¢) (3 points) Is [x] + “yl — [a+ yl=0 or Lve.y € 2.