Ch1 PP SetTheory - math.drexel.edujsteuber/spr17/PP/Ch1_PP_SetTheory.… · 1. Consider the...
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1 Chapter 1 Set Theory 1. Consider the matrices A = 0 −2 −6 8 −7 5 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ , ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − − = 2 9 3 10 1 1 4 11 B , ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = 10 3 1 2 C . a) Compute the matrix product AB . b) Compute the matrix product CB . c) Find the transpose of each matrix, i.e., T T T C B A , , . 2. Let { } 9 , , 2 , 1 … = U be the universal set and let A = x ( x mod 2) ≡ 1 { } and B = x x 2 ≤ 25 { } . List the elements of the set A ∩ B . 3. [1.2] List the elements of each set where 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) Show that A is not a subset of B = x ∈ N x is even { } . b) Show that A is a proper subset of C = 1,2, 3, …,8,9 { } . Consider the following sets for problems 5 and 6. Let { } 9 , , 2 , 1 … = U be the universal set and let { } 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) B A ∪ and B A ∩ b) C A ∪ and C A ∩ c) F D ∪ and F D ∩ 6. [1.5] Find: a) E D B A , , , b) D F E D A B B A − − − − , , , c) F E D C B A ⊕ ⊕ ⊕ , ,
Transcript of Ch1 PP SetTheory - math.drexel.edujsteuber/spr17/PP/Ch1_PP_SetTheory.… · 1. Consider the...
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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
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A = x (x mod 2) ≡1{ }and
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B = x x 2 ≤ 25{ }.Listtheelementsoftheset
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A∩ B .
3. [1.2]Listtheelementsofeachsetwhere
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N = 1,2,3,…{ } .a)
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A = x ∈ N 3 < x < 9{ }b)
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B = x ∈ N x is even, x <11{ }c)
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C = x ∈ N 4 + x = 3{ }
4. [1.3]Let
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A = 2,3,4,5{ }.a) Showthat
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A isnotasubsetof
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B = x ∈ N x is even{ } .b) Showthat
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A isapropersubsetof
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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 ⊕⊕⊕ ,,
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7. [1.27]Listtheelementsofthefollowingsetsiftheuniversalsetis
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U = a,b,c,…,y,z{ }. Furthermore, identify which of the sets, if any, are equal.
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A = x x is a vowel{ }
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B = x x is a letter in the word " little"{ }
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C = x x precedes f in the alphabet{ }
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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
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c) ( )BCA −∩
10. [1.7]Prove: ABAB ∩=− .Thus,thesetoperationofdifferencecanbewrittenintermsoftheoperationsofintersectionandcomplement.NOTE:Unionmeans“or”andintersectionmeans“and”.
11. [1.30]Let A and B beanysets.Prove:a)
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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
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A andasciencerequirement
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B.Apollof140sophomorestudentsshowsthat:
60completed
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A ,45completed
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B,20completedboth
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A and
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B. UseaVenndiagramtofindthenumberofstudentswhohavecompleted:
a) atleastoneof
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A and
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Bb) exactlyoneof
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A or
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Bc) neither
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A nor
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B
A
C
BU
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15. [5.23]Supposeamong32peoplewhosavepaperorbottles(orboth)forrecycling,thereare30whosavepaperand14whosavebottles.Findthenumberofpeoplewho:
a) savebothb) saveonlypaperc) saveonlybottles
16. [BRK1.2#15]Let
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A,
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B,and
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C befinitesetswith
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n A( ) = 6 ,
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n B( ) = 8 ,
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n C( ) = 6,
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n A∪ B∪C( ) =11,
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n A∩ B( ) = 3,
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n A∩C( ) = 2,and
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n B∩C( ) = 5 .Find
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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
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A = a,b,c,d{ } .
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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
