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Given the universal set U consisting of all integers between and including 1 and 20, and let the set A consist of all of the powers of 2 (integers of the form 2n where n is a whole number) within the universal set. Find the complement of the set A. Write your answer in proper set notation, for example {1,2,3,4,5,6}.

Respuesta :

jacob193

Answer:

[tex]\begin{aligned}\lbrace & \; 3,\, 5,\, 6,\, 7,\, 9,\, 10, \\ &\; 11,\, 12,\, 13,\, 14,\, 15,\, 17, \\ &\; 18,\, 19,\, 20\rbrace\end{aligned}[/tex].

Step-by-step explanation:

The integer powers of [tex]2[/tex] between [tex]1[/tex] and [tex]20[/tex] are:

  • [tex]2^{0} = 1[/tex].
  • [tex]2^{1} = 2[/tex].
  • [tex]2^{2} = 4[/tex].
  • [tex]2^{3} = 8[/tex].
  • [tex]2^{4} = 16[/tex].

Thus, [tex]A = \lbrace 1,\, 2,\, 4,\, 8,\, 16 \rbrace[/tex].

The complement of set [tex]A[/tex] (denoted as [tex]A^{C}[/tex]) includes all elements of the universal set [tex]U[/tex] that are not in set [tex]A\![/tex].

For example, the element [tex]1[/tex] is in [tex]U[/tex] and is also in [tex]A[/tex]. Thus, [tex]1[/tex] isn't in [tex]A^{C}[/tex].

The element [tex]3[/tex] is in [tex]U[/tex] and isn't in [tex]A[/tex]. Thus, [tex]3[/tex] is in [tex]A^{C}[/tex].

Repeat this decision process for each element of [tex]U[/tex] to enumerate [tex]A^{C}[/tex]:

[tex]\lbrace 3,\, 5,\, 6,\, 7,\, 9,\, 10, \, 11,\, 12,\, 13,\, 14,\, 15,\, 17, \, 18,\, 19,\, 20\rbrace[/tex].

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