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Chapter1SetTheory
1. Considerthematrices A =0 −2−6 8−7 5
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥, ⎥
⎦
⎤⎢⎣
⎡
−−
−=
2931011411
B , ⎥⎦
⎤⎢⎣
⎡=
10312
C .
a) Computethematrixproduct AB .b) ComputethematrixproductCB .c) Findthetransposeofeachmatrix,i.e., TTT CBA ,, .
2. Let { }9,,2,1 …=U betheuniversalsetandlet
€
A = x (x mod 2) ≡1{ }and
€
B = x x 2 ≤ 25{ }.Listtheelementsoftheset
€
A∩ B .
3. [1.2]Listtheelementsofeachsetwhere
€
N = 1,2,3,…{ } .a)
€
A = x ∈ N 3 < x < 9{ }b)
€
B = x ∈ N x is even, x <11{ }c)
€
C = x ∈ N 4 + x = 3{ }
4. [1.3]Let
€
A = 2,3,4,5{ }.a) Showthat
€
A isnotasubsetof
€
B = x ∈ N x is even{ } .b) Showthat
€
A isapropersubsetof
€
C = 1,2,3,…,8,9{ } .Considerthefollowingsetsforproblems5and6.Let { }9,,2,1 …=U betheuniversalsetandlet { }5,4,3,2,1=A { }9,8,7,6,5=C { }8,6,4,2=E { }7,6,5,4=B { }9,7,5,3,1=D { }9,5,1=F
5. [1.4]Find:a) BA∪ and BA∩ b) CA∪ and CA∩ c) FD∪ and FD∩
6. [1.5]Find:
a) EDBA ,,, b) DFEDABBA −−−− ,,, c) FEDCBA ⊕⊕⊕ ,,
2
7. [1.27]Listtheelementsofthefollowingsetsiftheuniversalsetis
€
U = a,b,c,…,y,z{ }. Furthermore, identify which of the sets, if any, are equal.
€
A = x x is a vowel{ }
€
B = x x is a letter in the word " little"{ }
€
C = x x precedes f in the alphabet{ }
€
D = x x is a letter in the word " title"{ }
8. [1.9]IllustrateDeMorgan’sLaw BABA ∩=∪ usingVenndiagrams.Alsoillustrate𝐴 ∩ 𝐵 = 𝐴 ∪ 𝐵.
9. [1.34]TheVenndiagramshowssets CBA ,, .Shadethefollowingsets:
a) ( )CBA ∪−
b) ( )CBA ∪∩
A
C
BU
A
C
BU
3
c) ( )BCA −∩
10. [1.7]Prove: ABAB ∩=− .Thus,thesetoperationofdifferencecanbewrittenintermsoftheoperationsofintersectionandcomplement.NOTE:Unionmeans“or”andintersectionmeans“and”.
11. [1.30]Let A and B beanysets.Prove:a)
€
A isthedisjointunionof BA − and BA∩ (i.e. ( ) ( )BABAA ∩∪−= ).b) BA∪ isthedisjointunionof BA − , BA∩ ,and AB − (i.e.
( ) ( ) ( )ABBABABA −∪∩∪−=∪ ).
12. [1.38]UsethelawsinTable1-1toproveeachidentity:a) 𝐴 ∩ 𝐵 ∪ 𝐴 ∩ 𝐵 = 𝐴b) 𝐴 ∪ 𝐵 = (𝐴 ∩ 𝐵) ∪ (𝐴 ∩ 𝐵) ∪ (𝐴 ∩ 𝐵)
13. [1.33]Theformula BABA ∩=− definesthedifferenceoperationintermsoftheoperationsofintersectionandcomplement.Findaformulathatdefinestheunion BA∪ intermsoftheoperationsofintersectionandcomplement.
14. [1.14]EachstudentinLiberalArtsatsomecollegehasamathematicsrequirement
€
A andasciencerequirement
€
B.Apollof140sophomorestudentsshowsthat:
60completed
€
A ,45completed
€
B,20completedboth
€
A and
€
B. UseaVenndiagramtofindthenumberofstudentswhohavecompleted:
a) atleastoneof
€
A and
€
Bb) exactlyoneof
€
A or
€
Bc) neither
€
A nor
€
B
A
C
BU
4
15. [5.23]Supposeamong32peoplewhosavepaperorbottles(orboth)forrecycling,thereare30whosavepaperand14whosavebottles.Findthenumberofpeoplewho:
a) savebothb) saveonlypaperc) saveonlybottles
16. [BRK1.2#15]Let
€
A,
€
B,and
€
C befinitesetswith
€
n A( ) = 6 ,
€
n B( ) = 8 ,
€
n C( ) = 6,
€
n A∪ B∪C( ) =11,
€
n A∩ B( ) = 3,
€
n A∩C( ) = 2,and
€
n B∩C( ) = 5 .Find
€
n A∩ B∩C( ) .
17. [BRK1.2#28]TheHansconductedasurveyoffreshmenonthefoodplanabouttheirpreferencesforfruits,vegetables,andcheese.Ofthe100freshmenquestioned,37saytheyeatfruits,33saytheyeatvegetables,9saytheyeatcheeseandfruits,12eatcheeseandvegetables,10eatfruitsandvegetables,12eatonlycheese,and3reporttheyeatallthreeofferings.Howmanypeoplesurveyedeatcheese?Howmanydonoteatanyoftheofferings?
18. [BRK1.2#26]Asurveyof500televisionwatchersproducedthefollowinginformation:285watchfootballgames,195watchhockeygames,115watchbasketballgames,45watchfootballandbasketballgames,70watchfootballandhockeygames,50watchhockeyandbasketballgames,and50donotwatchanyofthethreekindsofgames.
a) Howmanypeopleinthesurveywatchallthreekindsofgames?b) Howmanypeoplewatchexactlyoneofthesports?
19. [1.15]Inasurveyof120people,itwasfoundthat:
65readNewsweek 20readbothNewsweekandTime45readTime 25readbothNewsweekandFortune42readFortune 15readbothTimeandFortune8readallthree
a) Findthenumberofpeoplewhoreadatleastoneofthethreemagazines.
b) FillintheeightregionsoftheVenndiagramspecifyingthenumberineachregion.(NOTE:Itiseasiertodo part (b) first .)
c) Findthenumberofpeoplewhoreadexactlyonemagazine.
20. [1.18]Determinethepowerset ( )APower of
€
A = a,b,c,d{ } .
5
21. [1.37]Writethedualofeachequation:a) 𝐴 = 𝐵 ∩ 𝐴 ∪ 𝐴 ∩ 𝐵 b) 𝑈 = (𝐴 ∩ 𝐵) ∪ (𝐴 ∩ 𝐵) ∪ (𝐴 ∩ 𝐵) ∪ (𝐴 ∩ 𝐵)
22. [1.25]Provebyinduction:1+ 2! + 2! +⋯+ 2! = 2!!! − 1for𝑛 ≥ 0.
23. [1.50]Provebyinduction:2+ 4+ 6+⋯+ 2𝑛 = 𝑛(𝑛 + 1)for𝑛 ≥ 1.
24. [1.51]Provebyinduction:1+ 4+ 7+⋯+ 3𝑛 − 2 = !(!!!!)
!for𝑛 ≥ 1.
25. [BRK2.4#4]Provebyinduction:5+ 10+ 15+⋯+ 5𝑛 = !!(!!!)!
for𝑛 ≥ 1.
26. [1.52,BRK2.4#5]Provebyinduction:1! + 2! + 3! +⋯+ 𝑛! = !(!!!)(!!!!)!
for𝑛 ≥ 1.
27. [BRK2.4#2]Provebyinduction:1! + 3! + 5! +⋯+ (2𝑛 − 1)! = !(!!!!)(!!!!)
!for𝑛 ≥ 1.
Othersuggestedproblemsfromchapter1:1,11,29,41,42