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 TO RECEIVE FULL CREDIT!!! No. Score 1 /8 2 /8 3 /8 4 /8 5 /8 6 /8 7 /8 8 /8 9 /8 10 /8 11 / 10 12 /12 Bonus / 10 I Total I / 100 I
1. Express the following sets in set-builder notation, using the most condensed notation possible. (a) {April, August}..1.t,-ito J.\(#fl (b) {9, 10, 11, 12,...}
2. Express the following sets using the roster method. J. (, ~I) (a) T he set of the four seasons in a year. \Gf<l'j' funlii\4i',.f.jl, ""I\'ltr J].,7. t,*~1- (b) {x IXENand 6 <x<10} [1i ~ I 'l, IO} )
cj.f, -#C 3. Det ermine whether 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- (~lt o 4. Calculate the number of subsets of the given set, then list all the subsets. {I,II,III} tr~, fir"), inrf l r, 1L L fil, nrt fx, nr I fr,lr,jirj
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 the following sets. ).'.f'h (a) An B [~I~) I ~. ~,~5l) (b) B U C' C' -= ~q, ~I~) [~JC' = { Q'~'5'~D J.~~$"I (c) (A n C)' A() (::::-1>_----, ':: U ~ fc 1 b) c., d, t) f I ~ I ~ ~ ).~dh~ (d) (A' n B)u(A' nc') AI,,;: fbfc,d,lj.f~ ~ A'n~::~~~ AI ()C' ~ 9> Jb) U4:: l~bli..,.4, ~l~ (e) (A U B )' n C Ave, :: fa,l,,~,~j ~ raub) I ::: fcel,, +} kf a, t-, +Jnf~IL, ~Ie,+~ =5c, dit,.f f
6. Use t he Venn diagram to represent each set in roster form. J.4,. ~~ (a) A r--- --- _ y lj ~,3, 4, $')(, J 1,~} :z.4,1j:3s" (b) Au B ~ ' '{ I ~ ~1 (c) (B U C)' ;.4/it ~~ (d) An B [4 1 'i I ~D.1,~ / it: 4( (e) An B n C Lhll
.1.4rt 4Co 7. Construct a Venn diagram illustrating the 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
3.1,-:ifd!i 8. Express the quantified statement in an equivalent way, t hen write the negation of the quantified statement. (a) All whales are mammals. ervi~ '. lwe we Vlo ~ It s iw- W'( II'.O.f- ~~s. ~~ Ul~ ~' Sc~ ~s art f'lof ~~~~. ~. ((. 0\ (b) Some students are business majors. {vivjw- ', 'T4r( VJcis-ls td l~s+ th.j.. s~ wlo is ~ 'ovs»..tss ~~. ~4;"'~" tvo s~ off bimss ~0tS.
9. Let p, q, and r be the given simple statements. Write each compound statement in symbolic form. ~. ;11*3 (a) p: I'm leaving. q: You're staying. You're staying and I'm not leaving. ~. d-/tj\ (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 the temperature outside is freezing, then the heater is working or the house is not cold.
10. Let p, q, and r be the given simple statements. Write each symbolic statement in words. 1. ~rt~)(a) p: The heater is working. q: The house is warm. 3. ~ i~5~ (b) p: Romeo loves Juliet. q: Juliet loves Romeo. + C'J (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) -> r Ie ~ ~~ i S" ~ lbs\) CN Wt ~Y\ \6l.tJ s+v';j I +k... ~ ~ -n, ~ ~c1.
11. Determine whet her each statement is true or false. J.'l/-J-d."=> (a) Ralph E {Ralph, Alice, Trixie, Norton} Tn.Je. (b) Canada C {Mexico, United States, Canada} (c) {Ralph} C {Ralph, Alice, Trixie, Norton} True (d) {4} E {{4},{8}} Tr\Jt. (e) {I, 4} ~ {4, I} FtJs~
.1.s,"*" 4~ 12. A survey of 180 college students was t aken to determine participation in various campus activities. Forty-three students were in fraternities, 52 participated in campus sports, and 35 participated 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 three activities. Of those surveyed, (a) How many participated 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 tutorial 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 explanation as possible. ((A n B) U C')' = (A' n C) U (B' n C) ((Ane,)uc')' : (1\115)' n c ~, 11. MIJr~..,.'s ~>Js :: (A' UB') (1 t L, ~ ~j~is ~s ::- (A' (\c.,') U(B1nC) ~7 +..t 'Pi6tribUPW LAw DR u C' "::: I,t;),'},' ~1i~V(' -:: \I :1, ~,~ " l(mro)vc' Y=.~, ~11- A, ~ '.>' (.11)~ AI nc=(,/t s' -= f / 4 ;11~ BlnC:: 4 1 (A'()(}VCB'nc) ::L{lC.,l n... ~o ~ art ~lu1-kj ~ ~ SCt~ njj\ I)t\~ So itt AA QJe /wj.