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

€

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

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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 ⊕⊕⊕ ,,

2

7. [1.27]Listtheelementsofthefollowingsetsiftheuniversalsetis

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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{ }

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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

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)

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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

€

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