Fall 2004 - Popescu's Class - Exam 1

Fall 2004 - Popescu's Class - Exam 1 - Q50 LU'R OHS Exam 1...

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Unformatted text preview: Q50 LU'R OHS Exam 1, Mathematics 109 Name: Dr. Cristian D. Popescu Student ID: October 18, 2004 Section Number: Note: There are 3 problems on this exam. You willnot receive credit unless you show all your work. No books, calculators, notes or tables are permitted. 1. (40 points) (1) Use truth tables or other well known tautologies (eg. the de Morgan laws) to show that the following logical expression is a tautology. (PMQ) V (QM?) *—> (P V QMMP A Q) (2) Use the tautology in (1) above to show that if A and B are two subsets of a universal set M, then (A\B)LJ(B\A)=(AUB)\(ARB). (3) Compute (A\B)U(B\A),for A={x|:rEN, EkEZ, m=4k+1} B={m|xEN, SEEZ, x=2€+1}. Cl] Q M 03v LG! MP.) @QEIGGDVQ nfimm c—-—> @v m mum» A (Q v‘m A WW“) a £75 @VGA AUWW— HQ" (-9 34 .x e 'u ) Q00 A WQLQ \r (goo M?ch <—> <—-> 0(1) v ab.ka (Poe) A film) ® II. (30 points) (1) Write formally the following statement: “ For all real numbers I and y, such that x < —4 and y > 2, the distance between the point of coordinates (:12, y) and the point of coordinates (l, “2) is at least 6. [2) W'rite the formal negation of the statement in (1) above. (3) Prove or disprove the statement in (1) above and clearly indicate the method of proof used. ‘Hint: Please recall that the distanCe between P_(e, e) and Q(c,d) is given by the formula d(P, Q) = t/(a. — c)2 + (b — of)? ——_________._——t «7'4 «11 2C. 0) V—XGR)¥~36R) C%<-W)AR'3>2) AVG-3 t3 3 (-31) we [vu- III. (30 points) (1) Write formally the following statement: “There exists a prime number p, for which there exists an integer m, such that 33 + 3:2 + :r : p — 1.” (2) Prove or diSprove the statement in (1) above. ' w. é=X?QH\ swim}. va gie’g31-1XQZ) x3+xflx=rfL ...
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