The Universal Set, U, consists of the natural numbers from 20 to 60 incluive Define or describe in words the following three (3) sets: factors of 64, prime numbers, and multiples of 3.

Anonym 2021-08-09 Answered
The Universal Set, U, consists of the natural numbers from 20 to 60 incluive
a. Define or describe in words the following three (3) sets: factors of 64, prime numbers, and multiples of 3.
b. List the elements in each of your sets:
\(\displaystyle{A}={\lbrace}\)
\(\displaystyle{B}={\lbrace}\)
\(\displaystyle{C}={\lbrace}\)
c. Determine the probability of each of the following:
\(\displaystyle{I}.{P}{\left({C}\right)}\)
\(\displaystyle{I}{I}.{P}{\left({A}\cup{B}\right)}\)
\(\displaystyle{I}{I}{I}.{P}{\left({A}\cap{B}\cap{C}\right)}\)
\(\displaystyle{I}{V}.{P}{\left({B}{C}\right)}\)
\(\displaystyle{V}.{P}{\left({\left({A}{B}\right)}\cap{C}\right)}\)

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Expert Answer

escumantsu
Answered 2021-08-10 Author has 9249 answers
Step 1: Given,
The universal set U, consist of the natural numbers from 20 to 60 inclusive. We have to answer the following...
Step 2: Explanation
Solution (a).
If we have 3 sets,
\(\displaystyle{S}_{{{1}}}=\) factors of 64 i.e. 1,2,4,8,16,32,64
\(\displaystyle{S}_{{{2}}}=\) prime numbers
\(\displaystyle{S}_{{{3}}}=\) multiple of 3
Now describe the sets in words,
\(\displaystyle{S}_{{{1}}}={\left\lbrace{x}:{x}\right.}\) is a factor of 64 and \(\displaystyle{20}\leq{x}\leq{60}\rbrace\)
\(\displaystyle{S}_{{{2}}}={\left\lbrace{x}:{x}\right.}\) is prime and \(\displaystyle{20}\leq{x}\leq{60}\rbrace\)
\(\displaystyle{S}_{{{3}}}={\left\lbrace{x}:{x}\right.}\) is multiple of 3 and \(\displaystyle{20}\leq{x}\leq{60}\rbrace\)
Solution (b).
List of the elements
\(\displaystyle{S}_{{{1}}}={\left\lbrace{32}\right\rbrace}\)
\(\displaystyle{S}_{{{2}}}={\left\lbrace{23},{29},{31},{37},{41},{43},{47},{53},{59}\right\rbrace}\)
\(\displaystyle{S}_{{{3}}}={\left\lbrace{21},{24},{27},{30},\ldots,{57},{60}\right\rbrace}\)
Solution (c).
Determine the probability
Since we have from the b part, \(\displaystyle{S}_{{{1}}}\cup{S}_{{{2}}}={\left\lbrace{32},{23},{29},{31},{37},{41},{43},{47},{53},{59}\right\rbrace}\)
\(\displaystyle{S}_{{{1}}}\cap{S}_{{{2}}}\cap{S}_{{{3}}}=\phi\)
\(\displaystyle{I}.{P}{\left({S}_{{{3}}}\right)}={\frac{{{14}}}{{{40}}}}={\frac{{{7}}}{{{20}}}}\)
\(\displaystyle{I}{I}.{P}{\left({S}_{{{1}}}\cup{S}_{{{2}}}\right)}={\frac{{{10}}}{{{40}}}}={\frac{{{1}}}{{{4}}}}\)
\(\displaystyle{I}{I}{I}.{P}{\left({S}_{{{1}}}\cap{S}_{{{2}}}\cap{S}_{{{3}}}\right)}={\frac{{{0}}}{{{40}}}}={0}\)
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