MATH 211 Final, Fall 2003

Math 211 Final (Fall 2004 ) Time 2hrs Name: ID#: Circle your preblem solving section number below : Section 1 12:00 F Section 2 Problem 1 (a,b, c) / 18 1 (a, e)/10 1 (f, g) / 10 2/15 3/10 4/14 5,6 /16 7/10 TOTAL pkey i OF bhiney I a 1- Answer appropriately the following: (5 pts) a) Is the union of two transitive relations a transitive relation? pts (5 pts) b) Is the function f:JR — IR defined by f(x) =x’ +9 a one to one function? (5 pts ) c) Is Ip A qv (p A 9) logically equivalent to p? (Spts) 4) Let f:X >Y bea function. If f"(B)= {& © ¥ such that f(x) e B}, is f° (U(A))= 4 for any subset A of X2 , (Spts) —_e) Let A, BandCbesets. Is (4-C)N(C-B)=0? AMERICAN UNIVERSIT | Libhary | ce BRERET Q inateR AN CATV ENSIE nt n Let =|—|-|—| f €Z,Is I = 40,1 $? LIBRARY (5 pts ) f) Let f(x) Hi B orn s Im t } | OF BERET ee emer! (5 pts ) g) Is it true that “if 3 divides the product m.n, then 3 divides m orn”? 2- Let > = {0,1} and let s, denote the number of strings of length n (> 1)with an even number of 0’s. a) Calculate s,,s,,s,and s, . b) Let ¥y be the set of strings of length n on © Give the recursive definition of ,, then prove that s, =2s5,_,. n c) Solve the recurrence relation acters ane eee mae MEHKICAN ENTVERSID 6 U LIBRARY > OF BEIRUT a A i damentcas avis (10 pts) 3+ Let s,= Qn +1)+ Qn +3)4+ Qn+5)+ wt (4n-1) for 221. LIBKAR y a) Give the expressions and values for s,, = 1, 2,3 oF BEIRUT 5 é So en re Expression of s, Value of s, b) Use mathematical induction to prove that s, = 3n? AMENICAN EO NIVERSTD A LIBRARY (SPs) 4. Let $= §1,2,3, dy rere 2000}. oe meaner er terre a) What is the number of integers in S that are divisible by 9, 11 or 13.? (6 pts ) b) What is the number of integers in S that are divisible by 9, 1lor 13 but not by both 9 and 11? (8 pts ) 5. Nine students enter one elevator in the basement of College Hall. Each of these students will exit at either floor 1, 2, 3, 4, or 5. In how many ways can this happen? we ee eee JAANE AN ENE EEE, ; LIBKARY ae BEIRUT ee (8 pts) 6 Show that if s(n) = aus Jand t(n) = a? ) then ~ =o ) f AMELIGAN UNL EISES ft (10pts) 7+ Define the relation < on JR x IR by: L 1B RAR Y @y)s@w) if x+y? <2? 4w? > y <2 + - Show that (IR x IR, < ) is a poset. wor (x,y) = @,w)