MATH 211 Exam 2, Spring 2005
American University of Beirut · Discrete Structures · 6 pages
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Math 211
Quiz I (Spring 2005 ) Time: 75 min.
Name ;:
ID# :
Circle your problem solving section number below :
Section 1 11:00 F Section2 1:00F Section3: 1:00F
Probiem Grade
1,2 | 19
3,4,5 19
6,7 /12
8,9 /12
10 /8
TOTAL /50
(5 pts) 1. Determine whether the following compound proposition is a_ tautology.
(pr a)alan—w)}i> Leva)
(4 pts) 2. Consider the statement p:="Flowers do not grow if you do not water them”.
a. Give the contra-positive form of p.
b. Give the converse form of p.
2
. s
(5 pts) 3. Use De Moryan’s law to prove :
(40B)u C=(4 ACU BAC) forall subsets 4, B and C of some universal set U.
As 5 A a
(2 pts) 4 Show that 75 =1- 7, for all sets 4 where 7, is the charactcristic function of a
set F. | . | ~X, -h
Qpts) 5. Show that member(4A- B) = member(A) A not member(B) for all subsets 4 and B
of a universal set U.
(6 pts) 6. Show that f'(SUT)= f7'(S)U f(T), for all subsets S and 7 of the co-domain
i axe f'(8$ur)
safer gl |
xrePfe)V x« £7)
13 | or
(6 pts) 7, Show that | «| =n for all integers 7. Name whether your proof is direct or
“4 | 4}
indirect. a hk kK _
£ ma K A \ L Ay : A= KK + |
WK 4 QIK ¢
Liat GKrey
uyp 43>
SK LK - nae
GQ > a
(6ptsy 8 Use mathematical induction to show that : j
> (4n4-1)4 (4n 43) 4 (404 5) +... + (SH — 1 = 12n? for all 2 21
wy
bad) + [geah) vtheegh en Us) = tale
\
at ad
(Opts) 9, Show that V2 is an irrational. Specify what method of proof you are using and if
any state preliminary results (lemmas)?
CU maf) - ( Um +d) me m 49) 4e.0 4 [bad - )
5
10. Consider the function f :/R x /R— SIRx IR defined by /(x,y) = (x-2y,x+ 2y).
a. Is fone to one? Justify your answer ‘ apo a
;
‘
yoo 54, 3) 4- 4 J i).
Keo pleat 4 _ yo) 2 (% LY Pen
+ (bye ' aa
.
. Ca tye
" CeO) aetye) = X2 re)
(8 pts)
)
AL, 4 v)
b. Is fonto? Justify your answer
¢c. Is finvertible? Justify your answer