[PDF] Solutions Section 1.3. A. List

View - Download Solutions

Previous PDF

Next PDF

Solutions

Chapter 1 Exercises

Section 1.1

1. 3.

5.©x2R:x2AE3ªAE©¡p3,p3

7.

©x2R:x2Å5xAE¡6ªAE{¡

2,¡3}

9. 11. 13. {x2Z:j6xjÇ5}AE{0} 15. 17. {2,4,8,16,32,64...}AE{2x:x2N} 19. 21.
{0,1,4,9,16,25,36,...}AE©x2:x2Zª 23.

25.©...,18

,14 ,12 ,1,2,4,8,...ªAE{2n:n2Z}

27.©...,¡¼,¡¼2

,0,¼2 ,¼,3¼2 ,2¼,5¼2 ,...ªAEnk¼2 :k2Zo

29.j{{1},{2,{3,4}},;}jAE3

39.

¡2¡112

41.

¡2¡11243.

{(x,y):jxjAE2,y2[0,1]}¡3¡2¡1123

¡2¡112

45.

¡2¡112

241
47.

¡3¡2¡112349.

{(x,xÅy):x2R,y2Z}¡3¡2¡1123

¡3¡2¡1123

51.{(x,y)2R2: (y¡x)(yÅx)AE0}¡3¡2¡1123

¡3¡2¡1123

Section 1.2

1.SupposeAAE{1,2,3,4}andBAE{a,c}.

(d)B£BAE{(a,a),(a,c),(c,a),(c,c)} (e);£BAE{(a,b):a2;,b2B}AE;(There are no ordered pairs(a,b)witha2;.) (f)(A£B)£BAE (2,(a,a)),(2,(a,c)),(2,(c,a)),(2,(c,c)), (3,(a,a)),(3,(a,c)),(3,(c,a)),(3,(c,c)), 3. 5.

242Solutions

Sketch the following Cartesian products on thex-yplane. 9. {1,2,3}£{¡1,0,1}¡3¡2¡1123

¡2¡112

11.[0,1]£[0,1]¡3¡2¡1123

¡2¡112

13. {1,1.5,2}£[1,2]¡3¡2¡1123

¡2¡11215.

{1}£[0,1]¡3¡2¡1123

¡2¡112

17.N£Z¡3¡2¡1123

¡2¡112

¡2¡112

Section 1.3

A.List all the subsets of the following sets.

1.The subsets of{1,2,3,4}are:{},{1},{2},{3},{4},{1,2},{1,3},{1,4},{2,3},{2,4},

3.The subsets of{{R}}are:{}and{{R}}.

5.The subsets of{;}are{}and{;}.

7.The subsets of{R,{Q,N}}are{},{R},{{Q,N}},{R,{Q,N}}.

B.Write out the following sets by listing their elements between braces. C.Decide if the following statements are true or false.

13.R3µR3istruebecause any set is a subset of itself.

. This is true. (The even-numbered ones are both false. You have to explain why.) 243

Section 1.4

A.Find the indicated sets.

3.P({{;},5})AE{;,{{;}},{5},{{;},5}}

5.P(P({2}))AE{;,{;},{{2}},{;,{2}}}

{a},;),({a},{0}),({a},{1}),({a},{0,1}), {b},;),({b},{0}),({b},{1}),({b},{0,1}), 11. {XµP({1,2,3}):jXj·1}AE B.Suppose thatjAjAEmandjBjAEn. Find the following cardinalities.

13.jP(P(P(A)))jAE2³

2(2m)´

15.jP(A£B)jAE2mn

17.j{X2P(A):jXj·1}jAEmÅ1

19. jP(P(P(A£;)))jAEjP(P(P(;)))jAE4

Section 1.5

(a)A[BAE{1,3,4,5,6,7,8,9} (b)A\BAE{4,6} (c)A¡BAE{3,7,1,9} (d)A¡CAE{3,6,7,1,9} (e)B¡AAE{5,8}(f)A\CAE{4} (g)B\CAE{5,8,4} (h)B[CAE{5,6,8,4} (i)C¡BAE ;

3.SupposeAAE{0,1}andBAE{1,2}. Find:

(a)(A£B)\(B£B)AE{(1,1),(1,2)} (c)(A£B)¡(B£B)AE{(0,1),(0,2)} (d)(A\B)£AAE{(1,0),(1,1)} (e)(A£B)\BAE ;(f)P(A)\P(B)AE{;,{1}} (g)P(A)¡P(B)AE{{0},{0,1}} (h)P(A\B)AE{{},{1}}

244Solutions

5.Sketch the setsXAE[1,3]£[1,3]andYAE[2,4]£[2,4]on the planeR2. On separate

drawings, shade in the setsX[Y,X\Y,X¡YandY¡X. (Hint:XandYare Cartesian products of intervals. You may wish to review how you drew sets like[1,3]£[1,3]in the Section 1.2.)Y

XX[YX\YX¡YY¡X12341234

12341234

12341234

12341234

12341234

7. Sketch the setsXAE©(x,y)2R2:x2Åy2·1ªandYAE©(x,y)2R2:x¸0ªonR2. On separate drawings, shade in the setsX[Y,X\Y,X¡YandY¡X.XYX[YX\YX¡YY¡X¡2¡112

¡2¡112

¡2¡112

¡2¡112

¡2¡112

¡2¡112

¡2¡112

¡2¡112

¡2¡112

¡2¡112

9. The first statement is true. (A picture should convince you; draw one if necessary.) The second statement is false: Notice for instance that(0.5,0.5)is in the right-hand set, but not the left-hand set.

Section 1.6

1. SupposeAAE{4,3,6,7,1,9}andBAE{5,6,8,4}have universal setUAE{n2Z:0·n·10}. (a)AAE{0,2,5,8,10} (b)BAE{0,1,2,3,7,9,10} (c)A\AAE ; (d)A[AAE{0,1,2,3,4,5,6,7,8,9,10}AEU (e)A¡AAEA(f)A¡BAE{4,6} (g)A¡BAE{5,8} (h)A\BAE{5,8} (i)A\BAE{0,1,2,3,4,6,7,9,10} 3. Sketch the setXAE[1,3]£[1,2]on the planeR2. On separate drawings, shade in the setsX, andX\([0,2]£[0,3]).X X

X\([0,2]£[0,3])¡1123

¡1123

¡1123

¡1123

¡1123

¡1123

5. Sketch the setXAE©(x,y)2R2:1·x2Åy2·4ªon the planeR2. On a separate drawing, shade in the setX. 245
XX

123123

123123

Solution of 1.6, #5.A

A(shaded)U

Solution of 1.7, #1.

Section 1.7

1.Draw a Venn diagram forA. (Solution above right)

3.Draw a Venn diagram for(A¡B)\C.Scratch work is shown on the right. The

setA¡Bis indicated with vertical shading.

The setCis indicated with horizontal shad-

ing. The intersection ofA¡BandCis thus the overlapping region that is shaded with both vertical and horizontal lines. The final answer is drawn on the far right, where the set(A¡B)\Cis shaded in gray.AABBCC 5. Draw Venn diagrams forA[(B\C)and(A[B)\(A[C). Based on your drawings, do you thinkA[(B\C)AE(A[B)\(A[C)?

If you do the drawings carefully, you will find

that your Venn diagrams are the same for both

A[(B\C)and(A[B)\(A[C). Each looks as

illustrated on the right. Based on this, we are inclined to say that the equationA[(B\C)AE (A[B)\(A[C)holds for all setsA,BandC.ABC 7. Suppose setsAandBare in a universal setU. Draw Venn diagrams forA\B andA[B. Based on your drawings, do you think it"s true thatA\BAEA[B?

The diagrams forA\BandA[Blook exactly

alike. In either case the diagram is the shaded region illustrated on the right. Thus we would expect that the equationA\BAEA[Bis true for any setsAandB.ABU

9.Draw a Venn diagram for(A\B)¡C.ABC

11.The simplest answer is(B\C)¡A.

13.One answer is(A[B[C)¡(A\B\C).

246Solutions

Section 1.8

(a) 4[ iAE1A iAE{a,b,c,d,e,f,g,h}(b)4\ iAE1A iAE{a,b}

3.For eachn2N, letAnAE{0,1,2,3,...,n}.

(a)[ i2NA iAE{0}[N(b)\ i2NA iAE{0,1} 5. (a) i2N[i,iÅ1]AE[1,1)(b)\ i2N[i,iÅ1]AE; 7. (a) i2NR£[i,iÅ1]AE ; 9. (a)

X2P(N)XAEN(b)\

X2P(N)XAE ;

11.Yes, this is always true.

13.The first is true, the second is false.

Chapter 2 Exercises

Section 2.1Decide whether or not the following are statements. In the case of a statement, say if it is true or false.

1.Every real number is an even integer. (Statement, False)

3.Ifxandyare real numbers and5xAE5y, thenxAEy. (Statement, True)

5.SetsZandNare infinite. (Statement, True)

7. The derivative of any polynomial of degree5is a polynomial of degree6. (Statement, False)

9.cos(x)AE¡1

This is not a statement. It is an open sentence because whether it"s true or false depends on the value ofx.quotesdbs_dbs27.pdfusesText_33

[PDF] air d'opéra definition

[PDF] technique de lancer de poids

[PDF] lancer du poids exercices

[PDF] dimension du cercle de lancer de poids

[PDF] lancer de poids definition

[PDF] comment mesurer un lancer de poids

[PDF] aire de lancer de javelot

[PDF] traçage aire de lancer de poids

[PDF] etapes d'enseignement du lancer du poids

[PDF] air expiré definition

[PDF] calculer le volume de dioxygène

[PDF] contact presse air france

[PDF] service communication air france

[PDF] abonnement air france sky priority

[PDF] air france relation presse