MGF 1106 Exam #lA
Name:
~SWfr K~
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 TIllS EXAM IN ORDER T O R ECEIVE FULL CREDIT!!!
No.
Score
1
/8
/8
/8
/8
/8
/8
/8
/8
/8
/8
2
3
4
5
6
7
8
9
10
11
12
Bonus
I Total I
/ 10
/12
/ 10
/ 100 I
1. Express t he following sets in set-builder notation, using t he
most condensed notation possible.
(a) {April, August}
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J.\(#fl (b) {9, 10, 11, 12, . .. } 2. Express t he following sets using the roster method.
J . (, ~I) ( a) T he set of the four seasons in a year.
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(b)
{x IXENand 6 <x<10}
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cJ.f , -#C 3. Det ermine whet her t he following sets are equal, equivalent , both,
or neit her. Explain your answer.
A = {1,2,3,4,5}
B = {O, 1,2,3,4}
a - d-(~lto 4. Calculate t he number of subsets of the given set, then list all
the subsets.
{I,II,III}
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inrf
l r, 1L L f Il, nrt fx,nr I
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5. Let
U = {a, b, c, d, e, j, g, h}
A = {a,g, h}
B = { b,g, h}
C = {b,c, d, e,j}
Find each of t he following sets.
).'.f'H (a) An B
[~I~)
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(b) B U C'
C' -= ~q, ~I~)
[~JC' ={ Q'~'5'~ D
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(c) (A n C)'
A() (::::-1>_----,
¢':: U ~ fC b) c., d, t) f ~ ~ ~
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1
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(d) (A' n B )u(A' nC')
AI,,;: fbfC,d,lJ.f~ ~ A'n~::~~~ AI ()C' ~
..,.4, ~l~
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Jb) U 4:: l~bli (e) (A U B )' n C
Ave, :: fa,l, ,~,~J ~ rAUB) I
:::
fcel, €, +}
kf a, t-, +Jnf~ I L, ~ Ie, +~ =5c, dIt,.f f
6. Use t he Venn diagram to represent each set in roster form.
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(a) A
r - -- - --
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:z.4,1J:3S" (b) Au B
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(c) (B
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(d) An B
U
C)' [4 1'i I ~D
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(e) An B n C
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.1.4rt 4Co 7. Construct a Venn diagram illustrating t he given sets.
A = {a,e,h,i}
B = {b, c, e, j, h, i}
C = {e , j,g}
U = {a, b, c, d, e, j, g, h, i}
u
8. Express the quantified statement in an equivalent way, t hen
write t he negation of the quantified stat ement.
3.1,-:ifd!i (a) All whales are mammals.
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Vlo
art f'lof
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(b) Some students are business majors. {viVJw- ',
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off
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~0tS . ~ 'ovs»..tss ~~. 9. Let p , q, and r be the given simple statements. Write each
compound statement in symbolic form.
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(a) p: I'm leaving.
q: You're staying.
You're staying and I'm not leaving.
~. d-/t J\ (b) p: You are human.
q: You have feathers. Not being human is necessary for not having feathers. ~ .d.I-t(oD (c) p: The temperature outside is freezing.
q: The heater is working.
r: The house is cold.
If t he t emperature outside is freezing, t hen t he heater is
working or the house is not cold.
10. Let p, q, and r be t he given simple statements. Write each
symbolic statement in words.
1. ~rt~)(a) p: The heater is working.
q: T he house is warm.
3. ~ i~5~ (b) p: Romeo loves Juliet.
q: Juliet loves Romeo.
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(q V p)
~. J I-#-~1(c) p: The temperature is above 85°.
q: We finished studying.
r: We go to the beach.
(p V q)
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r
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i
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11. Determine whet her each statement is t rue or false.
J.'l/-J-d."=> (a) Ralph E {Ralph, Alice, Trixie, Norton}
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(b) Canada C {Mexico, United States, Canada}
(c) {Ralph} C {Ralph, Alice, Trixie, Norton}
True
(d) {4} E {{4}, {8}}
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(e) {I , 4}
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~
{4, I}
.1.s ,"*" 4~ 12. A survey of 180 college students was t aken t o determine partic­
ipation in various campus activities. Forty-three students were
in fraternities, 52 participated in campus sports, and 35 partic­
ipated in various campus tutorial programs. Thirteen students
participated in fraternities and sports, 14 in sports and tutorial
programs, and 12 in fraternities and tutorial programs. Five
students participated in all t hree activit ies. Of those surveyed,
(a) How many part icipated in only campus sports?
(b ) How many participated in fraternities and sports, but not
tutorial programs?
(c) How many participated in fraternities or sports, but not
t utorial programs?
(01
(d) How many participated in exactly one of these activities?
(e) How many participated in at least two of t hese activities?
(f) How many did not participate in any of t he three activities?
Bonus. Prove that the following sets are equal. Give as much ex­
planation as possible.
((A n B) U C')' = (A' n C) U (B' n C)
((Ane,)uc')' : (1\115)' nc ~, 11. MIJr~ ..,.'s ~>Js
:: (A' UB') (1 t
L,
~ ~j~IS ~s
::- (A' (\c.,') U(B1nC) ~7 +€..t
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u
,
A~
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C' ":: I,t;),'},'
AInc=(, /t
~1i~V (' -:: \ I :1, ~,~ "
s' -= f / 4 ;11~
l(Mro)VC' Y=.~, ~11-
BlnC:: 4 11­
(A'()(}VCB'n c) ::L{lC.,l
n...
~o ~ art ~lU1-kJ ~ ~ SCt~
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