MGF 1106 Exam #lB
ID# _ _ _________
HONOR CODE: On my honor, I have neither given nor received any aid on this
examination.
Signature: _ _ _ _ _ _ _ _ _ _ _ _ _ _ __
Instructions: Do all scratch work on the test itself. Make sure your final answers
are clearly labelled. Be sure to simplify all answers whenever possible. SHOW
ALL WORK ON THIS EXAM IN ORDER T O RECEIVE FULL CREDIT!!!
No.
Score
1
2
/8
/8
/8
/8
/8
/8
/8
/8
/8
/8
3
4
5
6
7
8
9
10
11
12
Bonus
Total
I
/ 10
/ 10
/12
/ 100 I
1. Express the following sets in set-builder notation, using t he
most condensed notation p ossible.
{9,lO, 11,12,... }
J.I ,#/b (b)
{April, August }
2. Express the following sets using the roster method.
;).1, -IJ./~ (a) T he set of mont hs of the year that have exactly 30 days.
~ .\,*,;)..~ (b) { x I x EN and 7 < x < II} }
~I \
(0,
II} .;2.1, it'4 3. Determine whether t he following sets are equal, equivalent, both,
or neit her. Explain your answer.
A
B
=
=
{I, 3, 5, 7, 9}
{2,4, 6,8, IO}
d . ? , ~"!>" 4. Calculate t he number of subsets of the given set, then list all
the subsets.
{t , a, b}
<P
*) l C4l, ) b}
f
I
~t /'~' f*,b~/ ~C4,b1
St , t,~1
5. Let
U = { 1,2,3,4,5,a,b,c, d ,e} A
= { 1,3,5,b, d } B
C
= {2, 3, 4,
=
b, c, d} {1, 2 ,a,d,e} Find each of t he following sets. u:: l C,~,C.,d.,t./.fJ j/~) A:: 1Q)~, '" , e, :: \ h, ~/\J c~\
l"t.,«, t,f}
c2.3,.lQ.(a) B n C HJ,dTI
J. .~~"i~ (b) Au C'
C' =l3,'1/S.. b,C ~
Aue,:: ll, ~/\{'S'J h,c.,d} :;l.31~5d-
~C/:: ~ a/~I\'~l (c) (An B )'
~ns=~3,I;"dJ
A(Vo:: f~l~l
(Arro)' ::fl,2,L(,S", Qlc,tl
01.4,*10 (d) (C' n A) u (C' n B' )
c'nA::f~,'S/Ll i B'-:~I,5")~tt}
c'nB'~~5}
;J.4 I #~y
c'=1a,j.~~ ::If ~'SJb II
~ut-::' l,2/~,\.f } c.,,,,Lld,e}
(~cY {\ A -;:
C'AA :::
C'f)
(C'{\ A)U (e' n~')
(e ) (B U C)' n A
('OUt)' ~ Js~
CAnbY': llt,b, c,d)t,f}
~(f, 'I~1) B'= Je ", d /f ,! j
~' -: ~~ 1
(e'n A)V((I,,~,)
'P>V(:: { 1:"C.,dJt..~~,,,,\
(B v2(=.
liill
::-~
led
(~vtlt{\A
::0
6. Use t he Venn diagram to represent each set in roster form .
.1.1{,-t3~
(a) B
f 4 ,SI C. , 't , Ie) If J
J
.J , ~ , J3('
(b) B
uc
f \.f 1S', t. ,\ t , '1 , 1°
.V~, ~31-
(c) (A U B )'
f ( ~II3}
.:2.~:tJ:. \to
(d) A n C f(,, ( l , ~} .2.4,::ttt(1.
(e) Au B uC
1
tl , IJ\
~.41~ l{s" 7. Construct a Venn diagram illustrating the given sets.
A
= {4, 5, 6, 8}
{1, 2, 4, 5, 6,7}
C = {3, 4, 7}
U= {1, 2, 3, 4, 5, 6, 7, 8, 9}
jg =
u
8. Express the quantified statement in an equivalent way, then
write the negation of the quantified statement .
.,.1 ,-#-30
(a) All journalists are writers. ~ '. 1'krt o.r~
d :~
r\o
~wc~~ J-o
~~s+s
Grl
Not-
c.rt
~~
:? .I,:#f3 3 (b) Some thieves are not criminals.
e1 ', No-\-
JJ
~ews
Grt
CX\~~ s.
"j'. All ~" M~ arr.. if;~'~ .
~f
wn4ers 9. Let p, q, and r be the given simple statements. Write each
compound st atement in symbolic form.
3.~,. "t (a) p: I'm leaving.
q: You 're staying.
I'm leaving and you 're not staying.
~.J ,#J'1
(b) p: You are human.
q : You have feathers. Having feathers is sufficient for not being human. (c) p: The t emperature outside is freezing.
q: T he heater is working.
r: T he house is cold.
If the temperat ure outside is freezing or t he heater is not
working, t hen t he house is cold.
10. Let p , q, and r be t he given simple statements. Write each
symbolic statement in words.
J.2(~:.31o
(a) p: T he heater is working.
q: The house is warm.
"'pVq
3 · J/~ (b) p: Romeo loves Juliet.
q: Juliet loves Romeo.
+
'" (p V q)
3.~'#-~ ~ (c) p: T he temperat ure is above 85°.
q: We finished studying. r' : We go to the beach. (q V r)
I}
Wt.
-+
p
h~~U ~~J
~~ I ~
a\ooV't
or ItJIo
~5 .
J>
n
i(,.
~ck, ~ 4..a.
11 . Determine whether each statement is true or false .
..?~ f *J~ (a) Canada E {Mexico, United States, Canada}
T~
J.. ~ 1f: ~S"
(b) Ralph C {Ralph, Alice, Trixie, Norton}
FrJ~
(c) {Canada} C {Mexico, United States, Canada}
Trvt..
(d) {I}
E
{{I}, {3}} Tnrt (e) {1 ,4} £ {4,1 }
TNt
J.s/:l:-lf(c, 12. A survey of 180 college students was t aken t o det ermine partic­
u.
ipation in various campus activities. Forty-three students were
in fraternit ies, 52 part icipat ed in campus sp orts, and 35 partic­
ipat ed in various campus t utorial programs. Thirteen students
participated in fraternities and sports, 14 in sports and t utorial
programs, and 12 in fraternities and t utorial programs. Five
students participated in all three activities. Of those sUl'veyed,
(a) How many participated in only campus sports?
3D
(b) How many part icipated in frat ernit ies and sports, but not
t ut orial programs?
(c) How many participated in fraternities or sports, but not
tutorial programs?
(d) How many part icipated in exactly one of t hese activities?
(e) How many participated in at least two of these activities?
(f) How many did not part icipate in any of the t hree activit ies?
Bonus. Prove that the following sets are equal. Give as much ex­
planation as possible.
(AI U (B
~I U (ifid)' ::
n G))' =
n B') u (A n G')
~ a. M"'j....·s ~.
A 11 (B(1C)'
:: A{\ (8' Ut')
(A
b7
lL MOijlWlfS LA\V~
~ (Ane')\J(A~C() ~7 tLt Dl~tr)bvhrl ~
A
~I ~ I/ ~ j"":l " M-c' ;:: I,l{
~
-tt.Jo
{"~ilMS
~
So
4/-t
"ttfrf~t d ~I ~ ~
it.. No AA
tAJ"f
i vcJ .