Answer to Question #171722 in Discrete Mathematics for Mary-Ann Asuncion

Question

A. Write each of the following sets by listing their elements between braces.

2. {3x+2:xϵZ}

4. {xϵN:-2<x≤7}

6. {xϵR:x²=9}

8. {xϵR:x³+5x²=-6x}

10. {xϵR:sin πx=0}

B. Write each of the following sets in set-builder notation.

18. {0,4,16,36,64,100,…}

20. {…,-8,-3,2,7,12,17,…}

22. {3,6,11,18,27,38,…}

24. {-4,-3,-2,-1,0,1,2}

26. {…,1/27,1/9,1/3,1,3,9,27,…}

C. Find the following cardinalities.

30. │{{1,4},a,b,{{3,4}},{Ø}}│

32. │{{{1,4},a,b,{{3,4}},{Ø}}}│

34. │{xϵN:│x│<10}│

36. │{xϵN:x²<10}│

38. │{xϵN:5x≤20}│

D. Sketch the following sets of points in the x-y plane.

40. {(x,y):xϵ[0,1],yϵ[1,2]}

42. {(x,y):x=2,yϵ[0,1]}

44. {(x,x²):xϵR}

46. {(x,y):x,yϵR,x²+y²≤1}

48. {(x,y):x,yϵR,x>1}

50. {(x,x²/y):xϵR,yϵN}

Answer 1

Solution.

A.

2. $\{3x+2:x\in Z\}=\{...,-4,-1,2,5,8,11,14,...\}.$

4.

$\{x\in N:-2<x\leq7\}=\{-1,0,1,2,3,4,5,6,7\}.$

6.

$\{x \in R:x^2=9\}=\{-3,3\}.$

8.

$\{x\in R:x^3+5x^2=-6x\}=\{-3,-2,0\}.$

10.

$\{x\in R:\sin{πx}=0\}=\{...,-3,-2,-1,0,1,2,3,...\}.$

B.

18.

$\{0,4,16,36,64,100,...\}=\{x\in Z:(2x)^2\}.$

20.

$\{...,-8,-3,2,7,12,17,...\}=\{x\in Z:5x-3\}.$

22.

$\{3,6,11,18,27,38,...\}=\{x\in N:x^2+2\}.$

24.$\{-4,-3,-2,-1,0,1,2\}=\{x \in Z:-4\leq x\leq 2\}.$

26.

$\{...,\frac{1}{27},\frac{1}{9},\frac{1}{3},1,3,9,27,...\}=\{x \in Z:3^x\}.$

C.

30.

$|\{\{1,4\},a,b,\{\{3,4\}\},\{\varnothing\}\}|=5.$

32.

$|\{\{\{1,4\},a,b,\{\{3,4\}\},\{\varnothing\}\}\}|=1.$

34.

$|\{x\in N:|x|<10\}|=9.$

36.

$|\{x\in N:x^2<10\}|=3.$

38.

$|\{x \in N:5x\leq 20\}|=4.$

D.

40.

42.

44.

46.

48.

50.