MATH 211 Final, Fall 2000

FINAL EXAM.; MATH 211 re FALL 2001. FEb. 3: 8:00 A.M.-10:00 A.M Ls Instructors’ Names: N. Fleihan A. Lyzzaik = lustructions: _ . 2 ¢ Make ste to write your NAME, ID NUMBER. and SECTION NUM- BER (or CLASS MEETING TIME) on your Examination book. e Write your answers on the colored EXAMINATION BOOKLET; re- serve an independent page to each independent question 1, 20565, 9, « The examination consists of 9 independent questions each of which cousists of partial questions, * The grade on each question is placed next to the question. * The TOTAL GRADE is 100. e N. P. and Z denote respectively the sets of natural numbers. positive integers, and integers. ¢ GOOD LUCK. +. Let the function f: Rx R= Rx R be defined by fey) = (2r3 + By? 3a? — 2y) (a) Show that f is a one-to-one correspondence, (6 points) (b) Find the inverse function of f. (2 points) 5. For mon € N define m~ n if m2 — n2 is a multiple of 5. (a) Show that ~ is an equivalence relation. (3 points) (b) Describe the equivalence classes, How tany equivalence classes are there? Justify your answer, (7 points) 6. (a) Let Ry and R, be relations on a set S. Show that (Rp 4 Ry)7 = Ri A ERg)O. {3 points) (b) Let Ry and Fy be relations ou a set S. Must Ry, aud Ry be transitive if Ry U Ro is transitive? Justify vour answer: If ¥es prove the statement, otherwise give a counter-example. (5 points) v. (a) Let f: 5S 3 T and g: TU be invertible finctions. Show that gof is invertible and (go f)~! = frto gu (5 points) (b) Prove that the sequence {1.5,---, 5"... ‘} has infinitely many terms such that the difference between any two of them is divisible by 11. Name your method of proof. (5 points) 3. fa) Show that nl" = Ofnl). (5 points) (b) Consider N x N and define the equivalence relation (m,n) ~ (p,q) if n= p. Determine the partition and the natural fimetion v of Nx N induced by ~ (5 points) 9. Answer by true (Tj or false (F) the following TEN questions. (2pts. each) GQ) EP Bis a relation on a set S. rhen RO RO is svinmetric. (i) A\ (BUC) =(A\ BYU(AL CO). (iii) AS(BEC)=(ASB EC. (iv) ye(n) = [n/2) — [n/2i.n © Z. where E is the set of even integers. (v) The equation +; 3 =x *5 2 has two solutions. (vi) There is only oue equivalence class for the equivalence relation in Z defined by m= 1 (mod 1). (vii) logy, 2 = O(log, i). (viii) Gn)! = On'h (ix) If sx, = 2s, +34 7n.n€P. then syn = O12"). (x) MATH 211 was FUN.