MATH 211 Final, Spring 2005

AM EHICAN ONIVER: LIBRARY = OF BEIRUT Final (Spring 2005 ) Time 2hrs Math 211 Circle your problem solving section number below : Miss Jaafar Section 1 11:00 F Section2 1:00F Miss Fuleihan Section3 1:00F Problem 2&3/12 | | 1/16 | 4/12 5/8 6&7/16 8/ 24 ( 16 pts ) ; . . \- Consider the set S = {2, 3,4, 6, 8. 24} . Define on S the relation “a/b if a divides b". a. Show that (S , |) is a poset. b. Draw the Hasse diagram of (S , |) ¢. Find the maximal and the minimal elements of (8 , | ) if they exist. d. Is(S,])a lattice? (4 pts) 2- Show that [(p q)a Gi r) (p r) is a tautology (8 pts ) 3- Use mathematical induction to show that 2 2. + On+l) _ (n+ fan+1X n+3) 4 i) 2 2 2 lr 43° 45° 4.. 4- Consider the function / : /Rx JR——*JRx IR defined by f(xy )= + 2y.2x-y). a. Show that / is one to one. b. Show that / is onto c. Determine the inverse function of f 5- On the set /N'x IN define the relation (m,n)=(p, g)if. m-3q = p-3n a. Show that ~ is an equivalence relation b. Describe the equivalence classes 6- How many integers from | through 1000 are (12 pts ) a. divisible by 3 or divisible by 7? b. divisible by 3 or divisible by 7 but not by both? ¢. not divisible by 3 or divisible by 7? (4 pts ) 7- Show thatif R is arelation ona set S,then RMR! is a symmetric relation on S. 8- Prove or disprove a AM(BUC)=(ANB)UC b. i 42n43nt1-0(") Prove or disprove a f(SuT)= s(S)Us) e. (2-1 )is divisible by 3 forall nEIN. (mod 4 ) nEZ 9- Prove that if 7 is an integer and 1° +5 is odd, then 7 is even usin g : g a. a direct proof b. an indirect proof c. aproof by contradiction